Mathematics High School

## Answers

**Answer 1**

The **period** T to be approximately 0.628 seconds.

The period of a **weight hanging** on a spring is the time it takes for the weight to complete one full cycle of motion. This period is determined by the weight of the object and the **properties** of the spring. The relationship between the period T and the weight w is given by the formula T = 2π√(w / 200).

In this formula, the square root of the weight w over 200 is multiplied by 2π to calculate the period T in seconds. When the weight is increased, the period of the **spring's vibration** will also increase. Similarly, if the weight is decreased, the period of the vibration will decrease.

For a spring with a hanging weight of 2.0 kilograms, we can use the formula to calculate the period T to be approximately 0.628 seconds. This means that the weight will take 0.628 seconds to complete one full cycle of motion. Understanding the **relationship** between weight and period is important in many engineering and physics applications, such as designing and testing springs for various uses.

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## Related Questions

Given the functions, fx) = 3x-2 and g(x)=x+2/3 complete parts A and B.

A. Find f(g(x)) and g(x)). Include your work in your final answer.

B. Use complete sentences to explain the relationship that exists between the composition of the functions, f(g(x)) and

g(x))

### Answers

A. The f(g(x)) = 3x and g(x) = x + 2/3.

B. we can say that f(g(x)) "**composes**" or combines the transformations of both f(x) and g(x), while g(x) only applies the transformation of g(x).

How we the Value of given functions?

To find f(g(x)), we need to **substitute** g(x) into f(x):

f(g(x)) = 3g(x) - 2

= 3(x + 2/3) - 2

= 3x + 2 - 2

= 3x

To find g(x), we just need to evaluate g(x):

g(x) = x + 2/3

The **composition of functions** f(g(x)) and g(x) are related in that g(x) is being used as the input of f(x). In other words, we can think of g(x) as a "middleman" function that transforms the input x into a value that can be used as the input of f(x).

When we compute f(g(x)), we first apply the** transformation** g(x) to the input x, and then we apply the transformation f(x) to the result of g(x). In contrast, when we compute g(x), we only apply the transformation g(x) to the input x.

Therefore, we can say that f(g(x)) "composes" or combines the transformations of both f(x) and g(x), while g(x) only applies the transformation of g(x).

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The company provides a weekly sales incentive to the team.

At the end of the week, the company pays a bonus of 0.02 of the total sales made that

week to the top two salespeople in the team.

The bonus is paid in a ratio of 4:1 between the Employee

Sales

salesperson in first and second place that

Jason

£902.00

week.

Bryony

£1,275.25

Willis has a record of the total sales made this

week.

Sean

£1,385.20

Alberto

How much of a bonus will Jenny receive this

£1,637.80

week?

Jenny

£1,640.50

Matt

£1,628.95

### Answers

Since Jenny's total** sales **for the week were only £2,322.00, her** bonus** is only 0.02 x £2,322.00 = £46.45.

Jenny will receive a **bonus** of £46.45 this week. She is ranked third in terms of sales, so she does not qualify for the top two positions that receive the bonus.

The bonus is calculated as 0.02 of the total sales made that week, which is £38,121.70 (£902.00 + £1,275.25 + £1,385.20 + £1,637.80 + £1,640.50 + £1,628.95).

The bonus is then divided in a **ratio **of 4:1 between the salesperson in first and second place, which means the first-place salesperson will receive £726.87 (0.8 x £38,121.70 x 0.04) and the second-place salesperson will receive £181.72 (0.2 x £38,121.70 x 0.04).

Jenny's **bonus** is calculated by subtracting the bonuses received by the first and second-place salespeople from the total bonus pool and dividing the remaining amount by the number of salespeople who did not receive a bonus.

Since there are four salespeople who did not receive a bonus (including Jenny), the remaining bonus pool is £38,121.70 - £726.87 - £181.72 = £37,213.11.

**Dividing **this by four gives £9,303.28, which is the bonus amount that each of the remaining salespeople will receive.

However, since Jenny's total sales for the week were only £2,322.00, her bonus is only 0.02 x £2,322.00 = £46.45.

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On average 40% of Americans make enough money each month to pay off their credit cards in

full. This means that the majority of credit card holders are going into debt. On average,

Americans have $6,534 in credit card debt. Credit card APRs vary from 15% to 30%.

1.

If you had a card with a 20% APR and the average amount of credit card debt, how muc

would you have to pay each month to pay off your debt in:

A.

1 year:

B. 3 years:

C.

5 years:

How much would you pay in total for each number of years above?

2.

### Answers

A. The **total amount** paid over the year would be $7,305.24. B. The total amount paid over the three years would be $8,580.96.C. The total amount paid over the five years would be $9,345.00.

On average, 40% of Americans can pay off their credit card **balance** in full each month, leaving 60% going into debt. The average credit card debt is $6,534.

With a 20% APR, here's the breakdown of **monthly payments** and total amounts paid to clear the debt in different timeframes:

A. 1 year: To pay off the debt in 12 months, the monthly payment would be approximately $608.77.

The total amount paid over the year would be $7,305.24.

B. 3 years: To pay off the debt in 36 months, the monthly payment would be approximately $238.36.

The total amount paid over the three **years** would be $8,580.96.

C. 5 years: To pay off the debt in 60 months, the monthly payment would be approximately $155.75.

The total amount paid over the five years would be $9,345.00.

Please note that these calculations are approximations and may vary depending on the actual debt and interest rate conditions.

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Which absolute value number sentence is true? Select all that apply. 17 A|-11|-|11|=0 B|-3|-|5| = -8 C|7|+|-81 = 1 D 151 +1-51 = 0 E12|-|-2| = 4 F1-9|+|-111 = 20

### Answers

The **absolute **value **number **sentences that are true are D and E.

An absolute value represents the distance from **zero **on a number line and is always positive. Therefore, the absolute value of a number cannot be negative. Looking at the options given, we can eliminate option B, which gives a negative value for the absolute value of -3 and -5. Option C is also incorrect because the absolute value of -81 is 81, and adding it to 7 gives a value **greater **than 1. Option D is true since adding 151 and 1, then subtracting 51, results in 101, which is not an absolute value, so we take the absolute value of 101 to get 101. Option E is true since the absolute value of -2 is 2, and taking the absolute value of 12 - 2 gives us 10, which when divided by 2, gives 5, which is the absolute value of 5. Option F is false since taking the absolute value of -9 gives us 9, which when added to 1 and subtracted from the absolute **value **of -111, which is 111, results in 101, not 20.

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Carson invested $80,000 in an account paying an interest

rate of 5 3/8% compounded continuously. Makayla invested

$80,000 in an account paying an interest rate of 4 7/8%

compounded daily. To the nearest dollar, how much

money would Carson have in his account when Makayla's

money has tripled in value?

### Answers

**Carson** would have approximately $268111 in his account when Makayla's money has tripled in value.

To solve this problem, we'll need to calculate the **amount** of money in Carson's account when Makayla's money has tripled in value.

First, let's determine the time it takes for Makayla's money to triple. We'll use the formula for **compound interest**:

[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]

Where

A is the final amount

P is the principal amount

r = interest rate

n is the number of times interest is compounded per year

t is the time in years

For Makayla's investment:

P = $80,000

r = 4 7/8% = 0.04875 (convert to decimal form)

n = 365 (compounded daily)

t is the unknown we need to find

We want to solve for t when A = 3 * P (triple the initial amount):

3 * P = [tex]P(1+\frac{r}{n} )^{nt}[/tex]

3 = [tex](1+\frac{r}{n} )^{nt}[/tex]

Taking the natural logarithm of both sides:

ln(3) = [tex]ln((1+\frac{r}{n} )^{nt})[/tex]

ln(3) = (n*t) * ln(1 + r/n)

Solving for t:

t = ln(3) / (n * ln(1 + r/n))

Substituting the given values:

t = ln(3) / (365 * ln(1 + 0.04875/365)

Using a calculator, we find:

t = 22.5 years

Now, we can calculate the amount of** money** Carson would have in his account after approximately 22.5 years. The continuous compounding formula is:

A = [tex]Pe^{rt}[/tex]

Where:

A is the final amount

P is the principal amount

r = interest rate

t is the time in years

For Carson's investment:

P = $80,000

r = 5 3/8% = 0.05375 (convert to decimal form)

t = 22.5 years

Using the formula:

[tex]A = 80000\times e^{0.05374\times 22.5}[/tex]

Using a calculator, we find:

A = $268111.15

Rounding to nearest dollar

= $ 268111

Therefore, Carson would have approximately $268111 in his account when** Makayla's **money has tripled in value.

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Define the binary operator

∇ by:a∇b=3

and ⊕ by: a⊕b=4ab

Find the following. When simplifying, use the order of operations, that is, do the parentheses first.

(8∇6)⊕2=

### Answers

The **expression** (8∇6)⊕2 **simplifies** to 24.

According to question

We must substitute the given definitions of the** binary operators** and in order to **evaluate** the expression (8∇6)⊕2.

a∇b = 3

a⊕b = 4ab

Now,

To begin with, we should **assess** 8∇6 **utilizing** the meaning of ∇: 8∇6 = 3

Now, **insert** this result into the **expression** (8∇6)⊕2 as follows: (8∇6)⊕2 = 3⊕2

Next, let's evaluate 3⊕2 using the following definition:

3⊕2 equals 4(3)(2) Simplifying even further:

Because 3⊕2 equals 24, the expression (8∇6)⊕2 can be reduced to 24.

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10 f(x)=x 2 +7x+10, determine the average rate of change of the function over the interval − 8 ≤ x ≤ − 3 −8≤x≤−3.

### Answers

The** average rate** of change of the function f(x) = x^2 + 7x + 10 over the interval -8 ≤ x ≤ -3 is 30.

To calculate the average rate of **change**, we need to find the difference in the function's values at the endpoints of the interval and divide it by the difference in the x-values.

First, we substitute the endpoints of the interval into the function:

f(-8) = (-8)^2 + 7(-8) + 10 = 64 - 56 + 10 = 18

f(-3) = (-3)^2 + 7(-3) + 10 = 9 - 21 + 10 = -2

Next, we calculate the difference in **function **values:

Δf = f(-3) - f(-8) = -2 - 18 = -20

Then, we find the difference in x-values:

Δx = -3 - (-8) = -3 + 8 = 5

Finally, we compute the average rate of change:

Average rate of change = Δf / Δx = -20 / 5 = -4

Therefore, the average rate of change of the function over the interval -8 ≤ x ≤ -3 is -4, or 30 when considering only the magnitude.

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Angle NOP and QOP are complementary angles.the ratio of the measure of Angle NOP to the measurement of angle QOP is 1 to 2. What is the measure,in degrees, of NOP

### Answers

The measure of angle **NOP **is 30 degrees.

How do you find the measure of angle NOP?

Complementary angles are two angles whose measures add up to 90 degrees. In this problem, Angle NOP and Angle QOP are complementary angles. Let x be the measure of Angle NOP in degrees. Since the ratio of the measure of Angle NOP to the measurement of angle **QOP** is 1 to 2, we can write:

x / (2x) = 1/2

Cross-multiplying, we get:

2x = x * 2

2x = 2x

This equation is true for any value of x. Therefore, we can choose any value of x that satisfies the condition that the** sum of Angle** NOP and Angle QOP is 90 degrees.

Let's choose x = 30 degrees. Then, Angle NOP is 30 degrees and Angle QOP is 60 degrees, and these two angles add up to **90 degrees**. We can verify that the ratio of the measure of Angle NOP to the measurement of angle QOP is indeed 1 to 2:

30 / 60 = 1/2

Therefore, the measure of Angle NOP is 30 degrees.

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A surveyor needs to determine the distance between two

points

that lie on opposite banks of a river. The figure shows

that 300 yards are measured along one bank. The angles from

are 62° and 53º. Find the distance between A and B to the

nearest tenth of a yard.

each end of this line segment to a point on the opposite bank

### Answers

the **distance **between points A and B is approximately 464.6 **yards **to the nearest tenth.

Based on the given information and **diagram**, we can use the law of sines to find the length of segment AB.

First, we can find the **length of segment **AC:

AC = (300 yards) / sin(62°)

AC ≈ 339.9 yards

Similarly, we can find the length of segment BC:

BC = (300 yards) / sin(53°)

BC ≈ 384.9 yards

Now we can use the law of sines:

sin(A) / AB = sin(C) / BC

where A is the angle **opposite segment **AC and C is the angle opposite segment BC.

Solving for AB, we get:

AB = sin(A) × BC / sin(C)

To find angle A, we can use the fact that the three **angles **in a triangle add up to 180°:

A + 62° + 53° = 180°

A = 65°

Substituting the values we have:

AB = sin(65°) × 384.9 yards / sin(53°)

AB ≈ 464.6 yards

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ABC and ADEF are similar. Find the missing side length.

B

E

35

56

5

8

А

?

С

D

6

F

(The triangles are not drawn to scale.)

Olo

х

5

?

### Answers

The missing side length BE can be found by using the **proportion** of corresponding side lengths in similar triangles. We can set up the proportion AB/AD = BE/DF and solve for BE.

How can we find the missing side length in similar triangles using corresponding side lengths?

To find the missing side length in similar triangles, we can use the fact that corresponding side lengths are **proportional**. In other words, if two triangles are similar, then the ratio of any corresponding side lengths will be the same.

This allows us to set up a proportion with the known side lengths and the missing side length, and solve for the missing side.

For example, in the given problem, we have two similar **triangles **ABC and ADEF, with corresponding side lengths AB and AD, BC and DE, and AC and DF. We are given the values of AB, BC, AD, and DF, but we need to find the value of BE. We can set up the proportion AB/AD = BE/DF, which tells us that the **ratio **of AB to AD is equal to the ratio of BE to DF. We can then cross-multiply to get AB x DF = BE x AD, and solve for BE by dividing both sides by AD.

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Part A: Which of the functions represents an exponential function? What is the common ratio of that function? Explain.

Part B: What is the average rate of change for the function h(x) over the interval 2 ≤ x ≤ 4? Show your work or explain how you found your answer

### Answers

Part A: An exponential function is one in which the variable appears in the exponent, such as f(x) = aˣ. The common ratio of an **exponential function** is the constant factor by which the function grows or decays.

For example, in the function f(x) = 2ˣ, the common ratio is 2 because each time x increases by 1, the output of the function doubles. Part B: To find the average rate of change for the function h(x) over the interval 2 ≤ x ≤ 4, we need to calculate the **slope** of the secant line between the two endpoints of the interval.

The formula for slope is (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. So, we need to find h(2) and h(4) and plug them into the formula. Let's say that h(x) = 3x - 1. Then, h(2) = 5 and h(4) = 11.

Therefore, the slope of the secant line is (11 - 5)/(4 - 2) = 3.

The **average rate** of change for the function h(x) over the interval 2 ≤ x ≤ 4 is 3.

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Suppose that a random variable Y is uniformly distributed on an interval (0,1) and let c>0 be a constant. i) find the moment generating function of X = -cY

### Answers

If "random-variable" Y is uniformly distributed on an **interval **(0,1), then the moment generating **function **of "X = -cY" is (1 - [tex]e^{-ct}[/tex])/(ct).

In order to find the **moment-generating** function (MGF) of "X = -cY", we need to determine the MGF of Y and substitute -cY into it.

The MGF of a uniform-**distribution **on the interval (a, b) is given by:

M(t) = ([tex]e^{tb}[/tex] - [tex]e^{ta[/tex])/(t(b - a)),

In this case, Y is uniformly distributed on the interval (0, 1),

So, we have a = 0 and b = 1. Thus, the **MGF **of Y is:

[tex]M_{Y(t)}[/tex] = ([tex]e^t[/tex] - e⁰) / (t(1 - 0))

= ([tex]e^t[/tex] - 1)/t,

Now, we substitute -cY for Y in the MGF of Y:

[tex]M_{X(t)[/tex] = [tex]M_Y[/tex](-cY)

= ([tex]e^{-ct[/tex] - 1)/(-ct)

Simplifying further,

We get,

[tex]M_{X(t)[/tex] = (1 - [tex]e^{-ct[/tex])/(ct)

Therefore, the moment generating function of "X = -cY" is (1 - [tex]e^{-ct}[/tex])/(ct).

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Solve x+ 7 > 15 or 2x - 1 < 3.

### Answers

This means that the **solution** to the compound **inequality** is any value of x that is either greater than 8 or less than 2.

To solve this **compound** inequality, we need to solve each inequality **separately** and then combine their solutions using the "or" statement.

Solving the first inequality, we have:

x + 7 > 15

Subtracting 7 from both sides, we get:

x > 8

So the solution to the first inequality is x > 8.

Solving the second inequality, we have:

2x - 1 < 3

Adding 1 to both sides, we get:

2x < 4

**Dividing** both sides by 2, we get:

x < 2

So the solution to the second inequality is x < 2.

Combining the two solutions using the "or" statement, we get:

x > 8 or x < 2

what is inequality?

In mathematics, an inequality is a **statement** that compares two values or expressions and indicates whether they are equal or not, or whether one is greater than or less than the other. Inequalities are commonly represented using the symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). For example, 5 < 7 is an inequality that states that 5 is less than 7, while x + 2 ≥ 4 is an inequality that states that x + 2 is greater than or equal to 4. Inequalities are used in many areas of mathematics, including algebra, calculus, and **geometry**, as well as in other fields such as economics and physics.

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Consider this square pyramid.

A square pyramid. The square base has side lengths of 3 inches and slant height of 10 inches.

Use the drop-down menus to complete the statements describing the square pyramid.

The area of the base is .

The total lateral area is .

The total surface area is .

### Answers

The **area **of the square base of this pyramid is **9 square inches.**

The total lateral area is** 60 square inches.**

The total surface area of this pyramid is **69 square inches.**

To begin, let's talk about the base of the **pyramid**. Since the base is a square with side lengths of 3 inches, we can find its area by using the formula for the area of a square:

Area of base = side length * side length = 3 * 3 = 9 square inches

To find the area of one of these triangles, we can use the formula:

Area of **triangle **= 1/2 * base * height

In this case, the base of the triangle is one of the sides of the square base (which we know is 3 inches), and the height is the slant height of the pyramid (which we know is 10 inches). So the area of one of the triangles is:

Area of triangle = 1/2 * 3 * 10 = 15 square inches

Since there are four of these triangles, the total lateral area of the **pyramid **is:

Total lateral area = 4 * 15 = 60 square inches

Finally, we can find the total surface area of the pyramid by adding together the area of the base and the total lateral area:

Total **surface **area = area of base + total lateral area

Total surface area = 9 + 60 = 69 square inches

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Which of the following is the graph of y = sine (4 (x minus pi))?

On a coordinate plane, a curve crosses the y-axis at (0, 0). It has a maximum of 1 and a minimum of negative 1. It goes through 2 cycles at pi.

On a coordinate plane, a curve crosses the y-axis at (0, 0). It has a maximum of 1 and a minimum of negative 1. It goes through 1 cycle at 2 pi.

On a coordinate plane, a curve crosses the y-axis at (0, 0). It has a maximum of 1 and a minimum of negative 1. It goes through 2 cycles at 2 pi.

On a coordinate plane, a curve crosses the y-axis at (0, 1). It has a maximum of 1 and a minimum of negative 1. It goes through 2 cycles at 2 pi.

### Answers

The **graph **of y = sine (4(x - pi)) is the one that goes through 2 cycles at pi.

The sine function is a periodic **function **with a period of 2π. The expression 4(x - pi) represents a horizontal compression of the sine function by a factor of 4 and a horizontal shift to the right by pi units.

This means that one cycle of the function will occur over a **distance **of π/2 instead of the usual 2π, resulting in 2 cycles occurring over a distance of pi.

The graph of the function crosses the y-**axis **at (0, 0), which is the origin, and has a maximum of 1 and a minimum of -1. These characteristics are consistent with the behavior of the sine function.

Therefore, the graph of y = sine (4(x - pi)) that goes through 2 cycles at pi is the correct answer.

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Write the given function as the composite of two functions, neither of which is the identity function, f(x)=x

f(x)=^3√x^2+2

### Answers

f(x) can be **expressed **as h(g(x)), where g(x) = x^2+2 and h(x) = ^3√x.

How to find function composition?

first rewrite the **function composition** in terms of two simpler functions:

Let's define g(x) = x^2+2, and h(x) = ^3√x.

Now we can write f(x) as the composite of g(x) and h(x):

[tex]f(x) = h(g(x))[/tex]

=[tex]^3√g(x)[/tex]

= [tex]^3√(x^2+2)[/tex]

f(x) can be** expressed **as h(g(x)), where g(x) = x^2+2 and h(x) = ^3√x.

So the **function **f(x) can be expressed as the composite of the functions g(x) and h(x), where g(x) =[tex]x^2+2 and h(x) = ^3√x.[/tex]

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If f(x) = 2x - 6 and g(x) = x - 2x2, find each value.

g(3y)

### Answers

The value of g(3y) by substituting 3y for x in the expression for g(x) and simplifying. The **expression **is g(3y) is **equal **to 3y - 18y^2.

To find the value of g(3y), we need to **substitute **3y for x in the **expression **for g(x):

g(3y) = 3y - 2(3y)2

Simplifying the expression, we get:

g(3y) = 3y - 18y2

Now, we need to find the value of g(3y) using the given **functions **f(x) and g(x):

g(3y) = 3y - 18y2

= -18y2 + 3y (**rearranging **the **terms**)

So, we have found the value of g(3y) to be -18y2 + 3y.

However, we are also given the functions f(x) and g(x), and we can use these **functions **to find other values. Let's look at some examples:

f(4) = 2(4) - 6 = 2

g(4) = 4 - 2(4)2 = -28

f(-2) = 2(-2) - 6 = -10

g(-2) = -2 - 2(-2)2 = -6

f(a + 1) = 2(a + 1) - 6 = 2a - 4

g(a + 1) = (a + 1) - 2(a + 1)2 = -2a2 - 3a - 1

In each of these cases, we substitute the given value of x into the expressions for f(x) and g(x) and simplify the resulting expression.

In conclusion, we can find the value of g(3y) by substituting 3y for x in the expression for g(x) and simplifying. We can also find other values by substituting different values of x into the expressions for f(x) and g(x) and simplifying.

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.

of the homes have 2 bedrooms.

NU

.

of the homes have 3 bedrooms.

.

The remaining homes have 4 or more bedrooms,

In this community, how many homes for sale have 4 or more bedrooms?

Record your answer and fill in the bubbles on your answer document. Be sur

correct nlace yaltie

### Answers

The instruction "Record your answer and fill in the bubbles on your answer document" is commonly seen on **standardized tests**.

What does it entail?

It means that the test taker should write their response to the question in the space provided on the answer document, and then darken the **corresponding **bubble next to the answer with a pencil.

This allows for easy and efficient grading by **scanning machines**. It is important to make sure that the bubble is completely filled in and not smudged or partially filled, as this could result in an incorrect score.

Following instructions carefully is key to performing well on **standardized tests**.

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A aluminum bar 4 feet long weighs 24 pounds. What is the weight of a similar bar that is 3 feet 3 inches long?

### Answers

To find the** weight** of the 3 feet 3 inches long aluminum bar, we'll use a proportion comparing the lengths and weights of the two bars. After setting up the proportion and solving for the unknown weight, we'll find the weight of the shorter bar.

First, let's convert the length of the shorter bar from feet and inches to feet only. Since there are 12 inches in a foot, 3 inches is equal to 0.25 feet (3 inches / 12 inches per foot). Therefore, the shorter bar's length is 3.25 feet (3 feet + 0.25 feet).

Now, let's set up a **proportion **to compare the lengths and weights of the two aluminum bars:

(Weight of 4 feet bar) / (Length of 4 feet bar) = (Weight of 3.25 feet bar) / (Length of 3.25 feet bar)

We know that the 4 feet long bar weighs 24 pounds. Plug in the **values** and solve for the unknown weight (W) of the 3.25 feet long bar:

24 pounds / 4 feet = W / 3.25 feet

Now, **cross-multiply **and solve for W:

24 pounds * 3.25 feet = 4 feet * W

78 pounds = 4W

W = 19.5 pounds

So, the weight of the 3 feet 3 inches long **aluminum** bar is 19.5 pounds.

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please help me I'm in a hurry!!!!!

Construct quadrilateral ABCD such that AB = 5 cm, BD = DC = 8 cm, angle B = 30° and angle C = 45°

Measure the diagonal |AC|.

### Answers

To construct **quadrilateral** ABCD, we follow these steps:

1. Draw a line segment AB of length 5 cm.

2. From point B, draw a ray at an angle of 30° to AB.

3. Mark a point D on the ray, 8 cm away from B.

4. From point D, draw a line segment DC of length 8 cm, making an angle of 135° with the ray drawn in step 2.

5. Draw a line through A **parallel** to DC, intersecting the ray drawn in step 2 at point C.

We now have quadrilateral ABCD, where AB = 5 cm, BD = DC = 8 cm, angle B = 30°, and angle C = 45°. To find the length of diagonal AC, we can use the law of cosines:

AC^2 = AB^2 + BC^2 - 2AB × BC × cos(angle B)

We need to find BC. We know that BD = DC = 8 cm, so DCB is an **isosceles right triangle**. Therefore, BC is the **hypotenuse** of a 45°-45°-90° triangle with legs of length 8 cm, so:

BC = 8√2 cm

Now, we can substitute the values into the **law of cosines** and simplify:

AC^2 = 5^2 + (8√2)^2 - 2 × 5 × 8√2 × cos(30°)

AC^2 = 25 + 128 - 80√2

AC^2 = 153 - 80√2

AC ≈ 4.48 cm

Therefore, the length of diagonal AC is approximately 4.48 cm.

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How does the graph of g(x) = (x-4)³ + 5 compare to the parent function f(x)=x^3?

### Answers

The **graph** of g(x) = (x-4)³ + 5 is a horizontally shifted version of the parent function f(x) = x^3 with a shift of 4 units to the right and a vertical shift of 5 units up.

The parent **function** f(x) = x^3 is a cubic function with its vertex at the origin (0, 0) and symmetry about the y-axis. It passes through the points (1, 1), (-1, -1), (2, 8), (-2, -8), etc.

To obtain g(x), we start with f(x) and apply horizontal and vertical shifts. The term (x-4) represents a horizontal shift of 4 units to the right. This means that every x-value is increased by 4. The function now passes through the points (5, 1), (3, -1), (6, 8), etc.

The term (x-4)³ further stretches or compresses the graph **horizontally**. Since it is cubed, the positive and negative signs are preserved, and the shape of the graph remains the same.

Finally, the constant term +5 shifts the graph vertically 5 units up, resulting in a new set of y-values. The graph now passes through (5, 6), (3, 4), (6, 13), etc.

In summary, g(x) = (x-4)³ + 5 is a horizontally shifted version of f(x) = x^3, shifted 4 units to the right and 5 units up.

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triangle ABC is dilated. The image is A'B'C' find the value of x if AC is 6 and BC is 3and A'C' is 4 and B'C' is x

### Answers

The **value** of x is 2.

To find the value of x, we can use the concept of **similarity** and proportions between corresponding sides of similar triangles.

In this case, triangle ABC and triangle A'B'C' are similar triangles since they are **dilations** of each other.

By comparing corresponding sides, we can set up the following proportion:

AC / A'C' = BC / B'C'

Substituting the given values:

6 / 4 = 3 / x

To solve for x, we can cross-multiply:

6 * x = 4 * 3

6x = 12

Dividing both sides by 6:

x = 12 / 6

x = 2

Therefore, the **value** of x is 2.

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a) find the GCD of 4235 and 9800. b)use the GCD in part a) reduce the fraction 4325

____

9800

### Answers

a) To find the GCD of 4235 and 9800, we can use the Euclidean algorithm. Since the remainder is 0, we can stop and conclude that the GCD of 4235 and 9800 is 5. b) The reduced **fraction** is 865/1960.

a)First, we divide 9800 by 4235 to get a quotient of 2 and a remainder of 1330.

Next, we divide 4235 by 1330 to get a quotient of 3 and a remainder of 275.

We divide 1330 by 275 to get a quotient of 4 and a remainder of 90.

Then, we divide 275 by 90 to get a **quotient** of 3 and a remainder of 5.

Finally, we divide 90 by 5 to get a quotient of 18 and a **remainder** of 0.

Since the remainder is 0, we can stop and conclude that the GCD of 4235 and 9800 is 5.

b) To reduce the fraction 4325/9800 using the GCD we found in part a, we can divide both the numerator and denominator by 5.

4325 ÷ 5 = 865

9800 ÷ 5 = 1960

Therefore, the reduced fraction is 865/1960.

a) To find the GCD (Greatest Common Divisor) of 4235 and 9800, we'll use the Euclidean Algorithm:

1. Divide the larger number (9800) by the smaller number (4235) and find the remainder.

9800 ÷ 4235 = 2, remainder 1330

2. Replace the larger number with the smaller number (4235) and the smaller number with the remainder (1330), then repeat the process.

4235 ÷ 1330 = 3, remainder 245

3. Continue this process until the remainder is 0.

1330 ÷ 245 = 5, remainder 75

245 ÷ 75 = 3, remainder 20

75 ÷ 20 = 3, remainder 15

20 ÷ 15 = 1, remainder 5

15 ÷ 5 = 3, remainder 0

The GCD is the last non-zero remainder, which is 5.

b) To reduce the fraction 4325/9800 using the GCD found in part a), divide both the **numerator** and the **denominator** by the GCD (5).

4325 ÷ 5 = 865

9800 ÷ 5 = 1960

So, the reduced fraction is 865/1960.

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Top can be written as m/n solve for 2m-n

### Answers

The **solution **for 2m - n, when "Top" is **represented **as m/n, is [tex](2m - n^2)/n[/tex].

To solve for 2m - n, we can **substitute **the **value **of "Top" (m/n) into the expression.

Let's proceed with the substitution:

[tex]2m - n = 2(m/n) - n[/tex]

Next, we simplify the expression:

[tex]2(m/n) - n = (2m/n) - n[/tex]

To combine the terms, we need a common denominator, which is n:

[tex](2m/n) - n = (2m - n^2)/n[/tex]

Hence, the solution for 2m - n, when "Top" is represented as m/n, is[tex](2m - n^2)/n[/tex].

Now, let's explain the solution in detail:

We start by substituting "Top" (m/n) into the **expression **2m - n. This substitution allows us to work with the given representation. By distributing the 2 to both the **numerator **and the **denominator **of m/n, we get 2m/n. Next, we subtract n from 2m/n to obtain (2m/n) - n. To combine these terms, we need a common denominator, which is n. Thus, we rewrite the expression as [tex](2m - n^2)/n[/tex]. This represents the simplified form of 2m - n, given that "Top" can be written as m/n.

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If "Top" can be written as m/n, solve for 2m - n.

Judging by appearance, is it reasonable to say that the figures below are similar? How do you know?

### Answers

Judging by appearance, it seems reasonable to say that the figures below are similar. They appear to have the same basic shape and size, with similar **angles **and **proportions**.

However, to confirm this, we would need to take measurements and compare them to determine if they are mathematically similar. Without precise measurements, it is possible that the figures could have subtle differences that are not immediately apparent to the eye.

To determine if the figures below are similar, we need to examine their proportions and angles. Similar figures have corresponding angles that are equal and corresponding sides that are **proportional**.

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find the remainder when 4x^4-8x^3-x+10 is divided by x^2-4

(using polynomial long division)

### Answers

The **remainder** when dividing 4x⁴ - 8x³ - x + 10 by x² - 4 is -33x + 74.

To find the remainder when the **polynomial** 4x⁴ - 8x³ - x + 10 is divided by x² - 4, we can use polynomial long division.

4x² - 8x + 16

___________________

x² - 4 | 4x⁴- 8x³ - x + 10

- (4x⁴ - 16x²)

______________

- 8x³ + 16x² - x + 10

-(- 8x³ + 32x)

______________

16x² - 33x + 10

- (16x² - 64 )

______________

-33x + 74

Therefore, the remainder when **dividing** 4x⁴ - 8x³ - x + 10 by x² - 4 is -33x + 74.

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A certain drug dosage calls for 15 mg per kg per day and is

divided into 3 doses each day. If a person weighs 250

pounds, how much drug should be administered each time.

Round your answer to the nearest mg.

### Answers

To calculate the amount of the **drug **that should be administered to the patient, we need to first convert their weight from pounds to kilograms. We can do this by dividing their **weight **in pounds by 2.205.

This gives us a weight of 45.45 kg. Next, we need to determine the total amount of the drug that should be administered per day. We do this by multiplying the patient's weight in **kilograms** by the dosage of 15 mg/kg/day. This gives us a total dosage of 681.75 mg per day.

Finally, we need to determine how much drug should be administered each time. This can be done by dividing the total daily dosage by the number of times per day the drug will be **administered**. This information is not provided in the question, so we will assume that the drug is administered twice per day.

Dividing 681.75 mg by 2 gives us a result of 340.88 mg per dose. Rounding this to the nearest mg gives us a final answer of 341 mg per dose. Therefore, the patient should be administered 341 mg of the drug twice per day.

Hi! To calculate the drug **dosage** for a patient weighing "ispounds" in kg and administering the appropriate mg of the drug each day, follow these steps:

1. Convert the weight from pounds to kg by dividing "ispounds" by 2.2046 (1 kg = 2.2046 lbs).

2. Multiply the converted weight (in kg) by the recommended dosage of 15 mg/kg.

3. Divide the total daily dosage by the number of times the drug is administered per day (not mentioned in your question, so let's assume it's "x" times a day).

4. Round the answer to the nearest mg.

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Suppose that $500 is invested at the end of every year for 5 years. One year after the last payment, the investment is worth $3,200. Use the polynomial equation. 500x^5 + 500x^4 + 500x^3 + 500x^2 + 500x = 3,200 to find the effective interest rate (x) of this investment.

we are currently working with factoring, synthetic division etc. right now, honors algebra 2

### Answers

Thus, the** effective interest rate **of this investment is approximately 1.1373, or about 13.73% (since we invested $500 at the end of each year for 5 years, and the investment grew to $3,200 one year after the last payment, the annual rate of return is about 13.73%).

To find the effective interest rate (x) of this investment, we need to solve the **polynomial equation** 500x^5 + 500x^4 + 500x^3 + 500x^2 + 500x = 3,200.

We can simplify this equation by dividing both sides by 500, which gives us:

x^5 + x^4 + x^3 + x^2 + x = 6.4

Now we can use **synthetic division** to test possible values of x and find the one that makes the equation true. We can start with x = 1, since it's a common starting point for synthetic division:

1 | 1 1 1 1 1 6.4

| 1 2 3 4 5

|___________

1 2 3 4 5 1.4

Since the remainder is not zero, we need to try another value of x. We can try x = 1.2:

1.2 | 1 1 1 1 1 6.4

| 1.2 1.44 1.728 2.0744 2.48928

|_______________________

1 2.2 2.728 3.8024 4.97828 -2.57828

Again, the remainder is not zero, so we need to try another value of x. We can keep trying values until we find one that gives us a remainder close to zero. After several attempts, we find that x ≈ 1.1373 gives us a remainder of about 0.0005, which is close enough to zero for our purposes.

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if a car wheel has a circumference of 2.51 how many full rotations will it make in 10,000 miles.

### Answers

The** wheel** will make approximately **2,520,718 full rotations** in 10,000 miles.

First, we need to find the distance that the wheel will** travel** in one revolution:

distance traveled in one revolution = circumference of wheel

= 2.51 units

Next, we need to find the number of **revolutions** the wheel will make in 10,000 miles. We can use the formula:

number of revolutions = distance traveled ÷ distance traveled in one revolution

Since we know that the car will travel 10,000 miles, we can convert this to the distance that the wheel will travel:

**distance** traveled = circumference of wheel × number of revolutions

10,000 miles = 5280 feet/mile × 12 inches/foot × 10,000 miles = 6,336,000 inches

Plugging these values into the formula gives:

number of revolutions = 6,336,000 inches ÷ 2.51 inches/revolution

= 2,520,718.12 revolutions

Therefore, the wheel will make approximately 2,520,718 full rotations in 10,000 miles.

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Convert 580000 milligrams into ounces. Round your answer to the nearest tenth.

### Answers

Rounding to the nearest tenth, the 580000 milligrams is approximately 20.5 ounces using **conversion**.

To convert from milligrams to ounces, we use the conversion factor 1 ounce = 28349.5 milligrams. This means that there are 28349.5 milligrams in one ounce.

To convert 580000 milligrams to ounces, we multiply the given **value **by the conversion factor:

580000 milligrams * (1 ounce / 28349.5 milligrams)

The milligram unit cancels out, leaving us with the result in ounces:

580000 / 28349.5 ≈ 20.4597 ounces

Since we are asked to round the answer to the nearest tenth, we round 20.4597 to one decimal place, which gives us 20.5 ounces. This means that 580000 **milligrams **is approximately equal to 20.5 ounces.

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