75 POINTS
Part A: The area of a square is (4x2 − 12x + 9) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points)

Part B: The area of a rectangle is (16x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)

The area is equal to 324π square units.

To find the area of a circle with radius 18 units, we can use the formula for the area of a circle, which is given by A = πr^(2), where r is the radius.

Given that the radius is 18 units, we can plug this value into the formula:

A = π(18)^2

Now, square the radius:

A = π(324)

Next, multiply 324 by π:

A = 324π

Therefore, the area of the circle with radius 18 units is 324π square units.

Now, let's compare the options with the given options:

a) 18π units^(2)- This is incorrect, as we calculated the area to be 324π square units.

b) 36π units^(2)- This is also incorrect, as our calculated area is 324π square units.

c) 9π units^(2)- This is incorrect, as our calculated area is 324π square units.

d) 324π units^(2)- This is the correct option , as we calculated the area to be 324π square units.

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Answer:

Option A: T(-3, -4)

Step-by-step explanation:

For a general point (x, y), if we apply the transformation:

T(a, b)

at that point, the new point that we will get is:

T(a,b)[ (x, y) ] = (x + a, y + b)

Notice that if w take the difference between the new point (x + a, y + b) and the original point (x, y)

we get:

(x + a, y + b) - (x, y) = (x + a - x, y + b  y) = (a, b)

These are x-value and y-value that describe our transformation T(a, b).

Then if we know that the original point is B (-2, 1)

and the transformed point is B'( -5, -3)

We can just take the difference to get:

(-5, -3) - (-2, 1) = (-5 - (-2), -3 - 1) = (-5 + 2, -4) = (-3, -4)

Then the transformation applied is:

T(-3, -4)

The correct option is A.

The solution to the system of equations is x = -3 and y = 2. The y-value of the solution is 2

To find the y-value of the solution to the system of equations, we can solve the system using any suitable method such as substitution or elimination.

Given system of equations:

3x + 5y = 1

7x + 4y = -13

Let's use the method of elimination to solve the system:

Multiply equation 1 by 4 and equation 2 by 5 to make the coefficients of y in both equations equal:

4(3x + 5y) = 4(1) --> 12x + 20y = 4

5(7x + 4y) = 5(-13) --> 35x + 20y = -65

Now, subtract equation 1 from equation 2 to eliminate the y term:

(35x + 20y) - (12x + 20y) = -65 - 4

35x - 12x = -69

23x = -69

x = -69/23

x = -3

Substitute the value of x into equation 1 to find y:

3(-3) + 5y = 1

-9 + 5y = 1

5y = 1 + 9

5y = 10

y = 10/5

y = 2

Therefore, the solution to the system of equations is x = -3 and y = 2. The y-value of the solution is 2.

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Answer: 4.25

Step-by-step explanation: Conversion a mixed number 3 1/

8

to a improper fraction: 3 1/8 = 3 1/

8

= 3 · 8 + 1/

8

= 24 + 1/

8

= 25/

8

To find new numerator:

a) Multiply the whole number 3 by the denominator 8. Whole number 3 equally 3 * 8/

8

= 24/

8

b) Add the answer from previous step 24 to the numerator 1. New numerator is 24 + 1 = 25

c) Write a previous answer (new numerator 25) over the denominator 8.

Three and one eighth is twenty-five eighths

Conversion a mixed number 2 3/

8

to a improper fraction: 2 3/8 = 2 3/

8

= 2 · 8 + 3/

8

= 16 + 3/

8

= 19/

8

To find new numerator:

a) Multiply the whole number 2 by the denominator 8. Whole number 2 equally 2 * 8/

8

= 16/

8

b) Add the answer from previous step 16 to the numerator 3. New numerator is 16 + 3 = 19

c) Write a previous answer (new numerator 19) over the denominator 8.

Two and three eighths is nineteen eighths

Add: 25/

8

+ 19/

8

= 25 + 19/

8

= 44/

8

= 4 · 11/

4 · 2

= 11/

2

For adding, subtracting, and comparing fractions, it is suitable to adjust both fractions to a common (equal, identical) denominator. The common denominator you can calculate as the least common multiple of both denominators - LCM(8, 8) = 8. In practice, it is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 8 × 8 = 64. In the next intermediate step, , cancel by a common factor of 4 gives 11/

2

.

In words - twenty-five eighths plus nineteen eighths = eleven halfs.

Conversion a mixed number 1 1/

4

to a improper fraction: 1 1/4 = 1 1/

4

= 1 · 4 + 1/

4

= 4 + 1/

4

= 5/

4

To find new numerator:

a) Multiply the whole number 1 by the denominator 4. Whole number 1 equally 1 * 4/

4

= 4/

4

b) Add the answer from previous step 4 to the numerator 1. New numerator is 4 + 1 = 5

c) Write a previous answer (new numerator 5) over the denominator 4.

One and one quarter is five quarters

Subtract: the result of step No. 3 - 5/

4

= 11/

2

- 5/

4

= 11 · 2/

2 · 2

- 5/

4

= 22/

4

- 5/

4

= 22 - 5/

4

= 17/

4

For adding, subtracting, and comparing fractions, it is suitable to adjust both fractions to a common (equal, identical) denominator. The common denominator you can calculate as the least common multiple of both denominators - LCM(2, 4) = 4. In practice, it is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 2 × 4 = 8. In the next intermediate step, the fraction result cannot be further simplified by canceling.

In words - eleven halfs minus five quarters = seventeen quarters.

Answer: SA = 2π(1.4)² + 2.8π(4.2) square feet

Step-by-step explanation:

formula for calculating surface is 2πr² + 2πr× height

To show that the total charge contained in the charge distribution is Q, we integrate the charge density over the entire volume. The charge density is given by:

ρ(r) = ρ₀(1 - r/R) for r ≤ R,

ρ(r) = 0 for r > R,

where ρ₀ = 3Q/πR³.

To find the total charge, we integrate ρ(r) over the volume:

Q = ∫ρ(r) dV,

where dV represents the volume element.

Since the charge density is spherically symmetric, we can express dV as dV = 4πr² dr, where r is the radial distance.

The integral becomes:

Q = ∫₀ᴿ ρ₀(1 - r/R) * 4πr² dr.

Evaluating this integral gives:

Q = ρ₀ * 4π * [r³/3 - r⁴/(4R)] from 0 to R.

Simplifying further, we get:

Q = ρ₀ * 4π * [(R³/3) - (R⁴/4R)].

Simplifying the expression inside the parentheses:

Q = ρ₀ * 4π * [(4R³/12) - (R³/4)].

Simplifying once more:

Q = ρ₀ * π * (R³ - R³/3),

Q = ρ₀ * π * (2R³/3),

Q = (3Q/πR³) * π * (2R³/3),

Q = 2Q.

Therefore, the total charge contained in the charge distribution is Q.

To show that the electric field in the region r > R is identical to that created by a point charge Q at r = 0, we can use Gauss's law. Since the charge distribution is spherically symmetric, the electric field outside the distribution can be obtained by considering a Gaussian surface of radius r > R.

By Gauss's law, the electric field through a closed surface is given by:

∮E · dA = (1/ε₀) * Qenc,

where ε₀ is the permittivity of free space, Qenc is the enclosed charge, and the integral is taken over the closed surface.

Since the charge distribution is spherically symmetric, the enclosed charge within the Gaussian surface of radius r is Qenc = Q.

For the Gaussian surface outside the distribution, the electric field is radially directed, and its magnitude is constant on the surface. Hence, E · dA = E * 4πr².

Plugging these values into Gauss's law:

E * 4πr² = (1/ε₀) * Q,

Simplifying:

E = Q / (4πε₀r²).

This is the same expression as the electric field created by a point charge Q at the origin (r = 0).

To derive an expression for the electric field in the region r ≤ R, we can again use Gauss's law. This time we consider a Gaussian surface inside the charge distribution, such that the entire charge Q is enclosed.

The enclosed charge within the Gaussian surface of radius r ≤ R is Qenc = Q.

By Gauss's law, we have:

∮E · dA = (1/ε₀) * Qenc.

Since the charge distribution is spherically symmetric, the electric field is radially directed, and its magnitude is constant on the Gaussian surface. Hence, E · dA = E * 4πr².

Plugging these values into Gauss's law:

E * 4πr² = (1/ε₀) * Q.

Simplifying:

E = Q / (4πε₀r²).

This expression represents the electric field inside the charge distribution for r ≤ R.

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Answer:

Step-by-step explanation:

Sorry I can't explain how it is done. It is very difficult to explain on paper.

Answer:

m = 0

Step-by-step explanation:

Solve: Add -35m to -5m

-5m = -35m

+35m

30m = 0 *note: its okay to have 0 in this spot*

divide 0/30

m=0 because you can't divide by 0

The value of m is 7 for the given statement -5m = -35.

The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.

Given an expression

-5m = -35

Divide by -5 on both sides

m = 7

Therefore, For the given expression -5m = -35 the value of m is 7.

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Answer:

Step 4

Step-by-step explanation:

Given: Steps used by Shelly to evaluate the expression

To find: step that includes Shelly's mistake

Solution:

According to PEMDAS,

P denotes Parentheses

E denotes Exponents

M denotes Multiplication

D denotes Division

A denotes Addition

S denotes subtraction

Step 3 is \(-20\div 5-1\) and step 4 is \(-20\div 4\)

In step 4, first subtraction operation has been applied before the division operation.

There is a mistake in step 4 as according to PEMDAS, first -20 should be divided by 5 then 1 should be subtracted from the resultant number.

Answer:

Step 4 is correct for plato

Step-by-step explanation:

-20 divided by 4.

Answer:

I believe it's 0

Step-by-step explanation:

if x=0 to the zero times anything would be zero

so if 2x squared would be

2x0 =

0

0squared

To calculate the probability of getting at most five successes, we can add up the probabilities of getting 0, 1, 2, 3, 4, or 5 successes

Sure, here are the expressions that can be used to compute each of the probabilities:

a) p(exactly five successes) = (7 choose 5) * p^5 * q^2
This expression uses the binomial coefficient (7 choose 5), which represents the number of ways to choose 5 successes out of 7 trials. We then multiply this by the probability of getting 5 successes (p^5) and the probability of getting 2 failures (q^2).

b) p(at least five successes) = p(five successes) + p(six successes) + p(seven successes)
To calculate the probability of getting at least five successes, we can add up the probabilities of getting exactly 5, 6, or 7 successes. We can use the expression from part a to calculate each of these probabilities.

c) p(at most five successes) = p(zero successes) + p(one success) + p(two successes) + p(three successes) + p(four successes) + p(five successes)
. We can again use the expression from part a to calculate each of these probabilities.

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The derivative of f(g(x)) with respect to x is  f'(1/x).

The term derivative refers to a type of financial contract whose value is dependent on an underlying asset, group of assets, or benchmark.

Given that, g(x)=ln x

Differentiation of ln x = 1/x

Hence, The derivative of f(g(x)) with respect to x is  f'(1/x).

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To find the height of the tree, we can use the tangent of the angle between the top of the tree and the tip of Ellie's shadow to find the height of the tree.

Tangent(theta) = Opposite / Adjacent

Opposite = height of tree

Adjacent = distance between Ellie and the tree

Tangent(theta) = height of tree / 16.2

We know that:

Tangent(theta) = 20.25 / 16.2

We can solve for the height of the tree by multiplying both sides of the equation by 16.2:

height of tree = Tangent(theta) * 16.2

height of tree = (20.25 / 16.2) * 16.2height of tree = 20.25

So, the height of the tree is 20.25 meters to the nearest hundredth of a meter.

Answer:  the height of the tree is 20.25 meters to the nearest hundredth of a meter.

Step-by-step explanation:

Answer:

z=16

Step-by-step explanation:

14 = z-2

2+14=z

16= z

Answer:

z = 16

Step-by-step explanation:

In order to answer this question we have to look at the equation 14 = z - 2 , and we see that z is the only variable in the equation so we are going to add 2 to -2 in order to get it off the side of the equal sign with the variable z and what we do to one side of the equal sign you do to the other so you are actually adding 2 to both -2 and 14.

14 = z - 2

+2       +2

16 = z

we got our answer pretty quickly so how do we know if 16 is equal to the variable z? well we are going to place 16 in the place of the variable z.

14 = z - 2  or  14 = (16) - 2

16 - 2 = 14

14 = 14

Hope this helps

(a) The approximated root of f(x) = e^x + x^2 - x - 4 is x ≈ 2.151586.

(b) The approximated root of f(x) = x^3 - x^2 - 10x + 7 is x ≈ -0.662460.

(c) The approximated root of f(x) = 1.05 - 1.04x + ln(x) is x ≈ -1.240567.

(a) Purpose: f(x) = ex + x2 - x - 4 To apply Newton's method, we must determine the function's derivative as follows: f'(x) = e^x + 2x - 1.

Now, we can use the formula to iterate: Choose an initial guess, x(0) = 0, and carry out the iterations as follows: x(n+1) = x(n) - f(x(n))/f'(x(n)).

1. Iteration:

Iteration 2: x(1) = 0 - (e0 + 02 - 0 - 4) / (e0 + 2*0 - 1) = -4 / (-1) = 4.

2.229280 Iteration 3: x(2) = 4 - (e4 + 42 - 4 - 4) / (e4 + 2*4 - 1)

x(3)  2.151613 The Fourth Iteration:

x(4)  2.151586 The Fifth Iteration:

x(5)  2.151586 The equation f(x) = ex + x2 - x - 4 has an approximate root of x  2.151586.

(b) Capability: f(x) = x3 - x2 - 10x + 7 The function's derivative is as follows: f'(x) = 3x^2 - 2x - 10.

Let's apply Newton's method with an initial guess of x(0) = 0:

1. Iteration:

x(1) = 0 - (0,3 - 0,2 - 100 + 7), or 7 / (-10)  -0.7 in Iteration 2.

x(2)  -0.662500 The Third Iteration:

x(3)  -0.662460 The fourth iteration:

The approximate root of the equation f(x) = x3 - x2 - 10x + 7 is x  -0.662460, which is x(4)  -0.662460.

c) Purpose: f(x) = 1.05 - 1.04x + ln(x) The function's derivative is as follows: f'(x) = -1.04 + 1/x.

Let's use Newton's method to make an initial guess, x(0) = 1, and choose:

z

1. Iteration:

x(1) = 1 - (1.05 - 1.04*1 + ln(1))/(- 1.04 + 1/1)

= 0.05/(- 0.04)

≈ -1.25

Cycle 2:

x(2) less than -1.240560 Iteration 3:

x(3) less than -1.240567 Iteration 4:

x(4)  -1.240567 The equation f(x) = 1.05 - 1.04x + ln(x) has an approximate root of x  -1.240567.

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The inequality that represents the minimum amount of money he should carry is  4x+5y  ≥ 52

What is inequality?

An inequality in mathematics depicts the connection between two values in an algebraic statement that are not equal. One of the two variables on the two sides of the inequality may be greater than, greater than or equal to, less than, or less than or equal to another value, according to inequality signals.

Let x be the  cost of each books

Let y be the cost of each notebook

Cost of 4 books = $32

Cost of  5 notebooks  = $20

Total cost = 32+20 = $52

At least shows the inequality ≥

So, the inequality that represents the minimum amount of money he should carry is

4x+5y  ≥ 52

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The definite integral that represents the area of the region enclosed by the polar curve r = 1 sin θ is ∫[a, b] 1/2 r^2 dθ

To determine the definite integral that represents the area of the region, we integrate the expression 1/2 r^2 with respect to θ over the interval [a, b], where a and b are the limits of the region.

In this case, the polar curve r = 1 sin θ represents a circle with radius 1 centered at the origin. As θ varies from 0 to π, the curve traces half of the circle in the positive direction. To find the area of the region enclosed by this curve, we integrate the expression 1/2 r^2 over this interval.

The expression 1/2 r^2 represents the area of a sector of the circle with radius r and central angle θ. Integrating this expression with respect to θ gives us the total area enclosed by the curve.

By evaluating the definite integral over the interval [a, b], we can find the area of the region enclosed by the polar curve r = 1 sin θ.

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50x30x9x70jssjgsvebdghxhdbgd

Answer:

Step-by-step explanation:

(5 * 10³) * (9*10⁷) = 5*9 * 10³⁺⁷

                          = 45 * 10¹⁰

                          = 4.5 * 10¹¹

(7*10⁵)÷ (2*10²) =

                         \(\frac{7}{2}*10^{5-2}\\\\=3.5*10^{3}\)

Step-by-step explanation:

If the data shows a leaner relationship between two variables, the line that best fits this linear relationship is known as a least-squares regression line, which minimizes the vertical distance from the data points to the regression line.

Pythagoras was an ancient Greek mathematician who founded the Pythagorean school of thought. The Pythagorean theorem is a fundamental concept in mathematics that is attributed to Pythagoras and his followers.

It states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. While this theorem is considered a cornerstone of mathematics today, it is important to understand that Pythagoras did not have access to the advanced mathematical tools and methods that we have today.

He had to rely on geometric constructions and reasoning to prove his theorem. Furthermore, Pythagoras believed that all numbers could be expressed as ratios of whole numbers, which is not always true in reality. Despite these limitations, the Pythagorean theorem has stood the test of time and continues to be a crucial tool in mathematics and other fields.

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Answer:

74

Step-by-step explanation:

All you have to do it first add f(x)+ g(x) and then plug in x=5.

(f+g) = f(x) + g(x)

= (7x- 2) + (9x-4)

= 7x - 2 + 9x - 4

= 16x - 6

(f + g)(5) = 16(5) - 6

= 80 - 6

= 74

Each friend will receive "x/5" antique letters, where "x" represents the total number of antique letters in the box.

Sasha wants to give an equal number of letters to each of her five friends. The box of antique letters contains the letters that she wants to distribute among her friends. The question is to determine how many antique letters each friend will receive.

The total number of antique letters in the box is unknown. Let's assume that there are "x" antique letters in the box. If Sasha wants to distribute "x" letters among five friends equally, she can find out how many letters each friend will receive by dividing the total number of antique letters by the number of friends. Hence, each friend will receive x/5 letters.

To summarize, each friend will receive "x/5" antique letters.

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Answer:

Head Jamz

Step-by-step explanation:

Noisy Music Regular Price $250 Clearance Special - 15% off

If 15% is off then we pay only 100-15 =85% from the original price.

$250 / 100 % = x / 85%

x= 250*85/100 = $ 212.5

Loud N Sharp Regular Price $200 $5 off for every $100 spent

If the price is 200 then will get $5+$5 =$10 discount

Final price will be 200-10 = $ 180

Music Mart Regular Price $300 Mail-in rebate for $115

$300-$115 = $ 185

Head Jamz Regular Price $225 Manager special – 25% off

If 25% is off we only pay 75% of the original price

$225/ 100% = x / 75%

x= 225*75/ 100 = $168.75 is the lowest price

Step-by-step explanation:i think its 23

Add the total percentage of households that have two or more televisions, then, calculate the number of households by multiplying 250 by the corresponding percentage. Then, find the total amount of households with less than two televisions and substract those quantities to find how many more are there with two or more televisions than those with less than two televisions.

The percentage of households with two televisions is 23%, with three televisions is 24%, for four televisions is 13% and 6% have five or more. Then, the total percentage of households with two or more televisions, is:

Since 66% of 250 households have two or more televisions, then the number of households in that category, is:

The percentage of households with less than two televisions is equal to:

The number of households with less than two televisions, is:

Therefore, the difference from households with two or more televisions and those with less than two televisions, is:

The, there are 80 more households that have two or more televisions than those with less than two televisions.

Answer:

The answer is 3%

Step-by-step explanation:

1.) $160 divided by $164.80 = 1.03

2.) 1.03 x 100 = 103

3.) 103 - 100 = 3

4.) 164.80 - 160 = 4.80

5.) 4.80 divided by 160 = 0.03

6.) 0.03 x 100 = 3%

Answer:

Step-by-step explanation:

Dilation is the transformation that changes the size of a shape.

4 is a positive number so the shape will increase in size.

So after dilating every side by 4:

0,1

-0.5,0

1,0

We get

0,4

-2,0

4,0