A flagpole casts a shadow 16 feet long.
Shanika is standing near the pole. Her
shadow is 4 feet long. Shanika's height is 5
feet. How tall is the flagpole?

The simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]

From the question, we are to simplify the expression

From the given information,

[tex]f(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30}[/tex]

and

[tex]g(x) =\frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]

Also,

[tex]R(x) = f(x) \div g(x)[/tex]

∴ [tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \div \frac{x^{2}-9x+20 }{x^{2}-12x+36}[/tex]

[tex]R(x) =\frac{x^{2}-11x+28 }{x^{2}-11x+30} \times \frac{x^{2}-12x+36 }{x^{2}-9x+20}[/tex]

Factoring each of the quadratics

[tex]R(x) =\frac{x^{2}-7x-4x+28 }{x^{2}-6x-5x+30} \times \frac{x^{2}-6x-6x+36 }{x^{2}-4x-5x+20}[/tex]

[tex]R(x) =\frac{x(x-7)-4(x-7) }{x(x-6)-5(x-6)} \times \frac{x(x-6)-6(x-6) }{x(x-4)-5(x-4)}[/tex]

[tex]R(x) =\frac{(x-4)(x-7)}{(x-5)(x-6)} \times \frac{(x-6)(x-6)}{(x-5)(x-4)}[/tex]

Simplifying

[tex]R(x) =\frac{(x-7)}{(x-5)} \times \frac{(x-6)}{(x-5)}[/tex]

[tex]R(x) =\frac{(x-7)(x-6)}{(x-5)(x-5)}[/tex]

[tex]R(x) =\frac{x^{2} -6x-7x+42}{x^{2} -5x-5x+25}[/tex]

[tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]

Hence, the simplified form of R(x) is [tex]R(x) =\frac{x^{2} -13x+42}{x^{2} -10x+25}[/tex]

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The total amount of Chandra's investments after 8 years is $110.03

The formula for calculating the compound interest is expressed as;

A = P(1+r/n)^nt

where

P is the amount invested = $75

r is the rate. = 0.048

t is the time =8years

n = 12 (monthly)

Substitute

A = 75(1+0.048/12)^96
A = 75(1.467)
A = 110.03

Hence the total amount of Chandra's investments after 8 years is $110.03

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V = π r2h. Volume Calculations. Round ( Cylinder) Tanks. Example 1. Find the Volume in cubic feet of a tank having a radius of 10 feet and ...

Missing: 7500m ‎Give ‎SF.

In the case of a hollow cylinder, we measure two radii, one for the inner circle and one for the outer circle formed by the base of a hollow cylinder. Suppose, r1 and r2 are the two radii of the given hollow cylinder with 'has the height, then the volume of this cylinder can be written as; V = πh(r12 – r22).

To find the volume of a box, simply multiply the length, width, and height — and you're good to go! For example, if a box is 5×7×2 cm, then the volume of a box is 70 cubic centimeters.

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The expression (quotient) which is modelled by the given diagram as in the task content is; (x²+x-2)/(x-1) = x +2.

It follows from the task content that the diagram model in discuss by observation involves a quotient in which case, the division is; (x-1).

Additionally, the dividend in the quotient by observation is the sum of terms as follows;

x² + x +x -x -1 -1 = x²+x -2.

Ultimately, the result of the division according to the model is; (x +2).

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Answer:

100 minutes

Step-by-step explanation:

3 hours= 180 min

180/9 = 20

20 x 5 =100 min

Vertical angles are always equal and congruent.

The information is incomplete. An overview will be given. It should be noted that vertical angles simply means the angles that are opposite each other when two lines cross.

Vertical angles are always equal and congruent. When two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.

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The function represented is analyzed below.

The function represented by the graph is related to a piecewise function.

The key features of the function include:

The function has a constant value in interval (0, 15) which is 40.

It decreases at first and then increases from 45 to 65.

It increases linearly from 65 to 90.

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Answer:

96

Step-by-step explanation:

(2^22-2^20) (3^24-3^22) (4^26-4^24)

=[(2^2-1)2^20][(3^3-1)3^22][(4^2-1)4^24]

=3*2^20*8*3^22*15*4^24

=2^23*3^23*4^24*15

=2^23*3^23*2^48*3*5

=2^71*3^24*5.

∵(2^22-2^20) (3^24-3^22) (4^26-4^24)=2^x × 3^y × 5^z，

∴x=71, y=24, z=1,

∴x+y+z=71+24+1=96.

The slope-intercept equation for the given line is y = 2x + 3

From the question, we are to determine the equation for the line

The slope-intercept form of the equation of a straight line is

y = mx + b

Where m is the slope

and b is the y-intercept

First, we will determine the slope

Slope of the line = (3 - 0) / (1.5 - 0)

Slope of the line = 3/1.5

Slope of the line = 2

∴ m = 2

On the graph, we can observe that the y-intercept is 3

∴ b = 3

Thus,

The equation becomes

y = 2x + 3

Hence, the slope-intercept equation for the given line is y = 2x + 3

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Answer:

x² + 5x - 14 = (x + 7)(x - 2)

x² - 3x - 54 = (x - 9)(x + 6)

x² + 8x + 12 = (x + 2)(x + 6)

x² - 11x + 30 = (x - 5)(x - 6)

x² - 25 = (x + 5)(x - 5)

x² + 8x - 9 = (x - 1)(x + 9)

Step-by-step explanation:

To factor x² + ax + b, look for two numbers whose sum is a and whose product is b.

The factorization of x² - a² is (x + a)(x - a).

x² + 5x - 14 = (x + 7)(x - 2)

x² - 3x - 54 = (x - 9)(x + 6)

x² + 8x + 12 = (x + 2)(x + 6)

x² - 11x + 30 = (x - 5)(x - 6)

x² - 25 = (x + 5)(x - 5)

x² + 8x - 9 = (x - 1)(x + 9)

The factored form of the given expressions are

y = (x -2)(x +7)

y = (x +6)(x -9)

y = (x +2)(x +6)

y = (x -5)(x -6)

y = (x +5)(x -5)

y = (x -1)(x +9)

y = (x +4)(x -4)

From the question we are to factor the given quadratic expressions

y = x² +7x -2x - 14

y = x(x +7) -2(x +7)

y = (x -2)(x +7)

y = x² -9x +6x - 54

y = x(x -9) +6(x -9)

y = (x +6)(x -9)

y = x² +6x +2x +12

y = x(x +6) +2(x +6)

y = (x +2)(x +6)

y = x² -6x -5x +30

y = x(x -6) -5(x -6)

y = (x -5)(x -6)

y = (x +5)(x -5)

y = x² +9x -x -9

y = x(x +9) -1(x +9)

y = (x -1)(x +9)

y = (x +4)(x -4)

Hence, the factored form of the given expressions are

y = (x -2)(x +7)

y = (x +6)(x -9)

y = (x +2)(x +6)

y = (x -5)(x -6)

y = (x +5)(x -5)

y = (x -1)(x +9)

y = (x +4)(x -4)

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Answer:

x = 0 , x = 5

Step-by-step explanation:

2x² - 10x = 0 ← factor out 2x from each term on the left side

2x(x - 5) = 0

equate each factor to zero and solve for x

2x = 0 ⇒ x = 0

x - 5 = 0 ⇒ x = 5

The length of the other parallel side is 3x

The given parameters are:

Midsegment= 2x

One parallel side = x

The midsegment is calculated as:

Midsegment = 0.5 * (sum of parallel sides)

So, we have

2x = 0.5 *(x + Other side)

Divide by 0.5

4x = x + Other side

Evaluate the like terms

Other side = 3x

Hence, the length of the other parallel side is 3x

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Here, given

sides of polygon= a(3,7)b(3,3)c(7,3)d(7,7)

Let, 3 = x1 & 7= y1   , 3= x2  & 7= y2   , 7= x3  & 3= y3   ,  7=  x4  & 7= y4

[tex]\frac{1}{2}[/tex] [det.(9-21)+det.(9-21)+det.(49-21)+det.(49-21)] {we took determinants of all the given sides}

[tex]\frac{1}{2}[/tex] (32)

16cm^2

Hence, the area of the polygon is 16cm2.

The formula for calculating the area of ​​a regular polygon is: area = (number of sides x 1 side length x apothem distance) / 2, where the value of the apothem is the formula apothem = [(1) Length side). / {2 × (tan (180 / number of sides))}].

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The slope-intercept formula of the linear equation is:

[tex]y = \frac{3}{7}x + 7[/tex]

A linear function is modeled by:

y = mx + b

In which:

The y-intercept is of b = 7. The line also goes through point (0,-3), hence the slope is:

m = 3/7.

Thus the equation is:

[tex]y = \frac{3}{7}x + 7[/tex]

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The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.

According to the statement

we have given that If 15 actuators have failed and we have to find the probabilities on some conditions.

we know that the formula of probabilities is

probability = possible outcomes / total outcomes

So,

at least 10 are repairable = 1 - 10/15

at least 10 are repairable = (15 - 10)/15

at least 10 are repairable = (5)/15

at least 10 are repairable = 1/3

from 3 to 8 are repairable = 1/5 *8/15

exactly 5 are repairable = 1/3

These are the probabilities of the given conditions.

So, The probabilities of at least 10 are repairable is 1/3. and probabilities of from 3 to 8 are repairable is 1/5*8/15 and probabilities of exactly 5 are repairable is 1/3.

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Answer:

[tex]\sf \dfrac{\sqrt{6}+\sqrt{2}}{4}[/tex]

Step-by-step explanation:

   75 = 45 + 30

Sin (x + y) = Sin x *Cos y + Cos x * Sin y

Sin (45 + 30) = Sin 45 * Cos 30 + Cos 45 * Sin 30

                      [tex]\sf =\dfrac{1}{\sqrt{2}}*\dfrac{\sqrt{3}}{2}+\dfrac{1}{\sqrt{2}}* \dfrac{1}{2}\\\\\\=\dfrac{\sqrt{3}}{2\sqrt{2}}+\dfrac{1}{2\sqrt{2}}\\\\\\=\dfrac{\sqrt{3}*\sqrt{2}}{2*\sqrt{2}*\sqrt{2}}+\dfrac{1*\sqrt{2}}{2\sqrt{2}*\sqrt{2}}\\\\\\=\dfrac{\sqrt{6}}{2*2}+\dfrac{\sqrt{2}}{2*2}\\\\\\=\dfrac{\sqrt{6}+\sqrt{2}}{4}[/tex]

The percentage that the number of airlines with an average airplane age of 14 is less than the number of airlines with an average airplane age of 11 is:

B. 20%.

The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:

[tex]P = \frac{a}{b} \times 100\%[/tex]

The percentage of airlines with an average age of 14 years is:

[tex]P = \frac{2}{15} \times 100\% = 13.33\%[/tex]

The percentage of airlines with an average age of 11 years is:

[tex]P = \frac{5}{15} \times 100\% = 33.33\%[/tex]

The difference is:

33.33% - 13.33% = 20%.

Hence option B is correct.

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The single transformation which takes shape A to shape B as in the task content is a 90° clockwise rotation and a 6 units shift rightward.

By observation, it follows that the orientation of the shape A differs from that of B as it lags by 90° in the clockwise direction.

Additionally, the resulting image from a 90° clockwise rotation about the left vertice of A still reguires that A be shifted 6 units rightwards to yield shape B.

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The rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet above the ground is  1/20 degree/sec.

It is described as a relationship between an oblique line from the observer's eye to an item above his eye and the horizontal plane. This angle eventually develops above the surface. The angle of elevation is made in such a way that it is above the observer's eye, as the name suggests.

Calculation for the rate of change of angle of elevation-

According to the question,

A balloon rises at the rate of 10 ft/sec from a point on the ground 100 feet from an observer.

The balloon is 100 feet above the ground.

Let the height of the balloon be 'h' ft.

Then tanØ = h/100

h = 100tanØ

Differentiate the above equation with respect to time.

dh/dt = 100 sec² Ø (dØ/dt)

We know dh/dt = 10 ft/sec.

when h = 100

tanØ = 100/100 = 1/1

tanØ = Perpendicular/Base

On comparing above two equations,

Perpendicular = 1; Base = 1.

By Pythagoras the hypotenuse is,

H² = P² + B²

    = 1² + 1²

    = 2

H = √2

Now, calculate the value of cosØ.

cosØ = B/H = 1/√2

Squaring both side

cos² Ø = 1/2

sec² Ø = 1/cos² Ø

sec² Ø = 2

The main equation was,

dh/dt = 100 sec² Ø (dØ/dt)

Substitute the values,

dØ/dt = (dh/dt )/100 sec² Ø

         = (10)/(100×2)

dØ/dt = 1/20 degree/sec.

Therefore, the rate of change of the angle of elevation of the balloon from the observer is  1/20 degree/sec.

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The exponential equation that represent the graph is y = 2(4)^x

The standard exponential function is expressed as

y = ab^x

Given the coordinate points (1, 8) and (0,2), the equivalent equations are;

8 = ab^1

2 = ab^0

from equation 2, a = 2

8 = 2b

b = 4

Hence the exponential equation that represent the graph is y = 2(4)^x

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The question that could be answered by using a logarithm is explained below

The question is: Let your dependent variable in the function be y. Write the function that models the independent variable in terms of y, using logarithms.

The question is asking you to solve y = f(x)

You already did this using a constant for y. Do the same thing with y instead of the constant.

y = 4(1.0256^x)

y/4 = 1.0256^x . . . . . . . divide by 4

log(y/4) = x·log(1.0256) . . . . . take logs

log(y/4)/log(1.0256) = x . . . . . divide by the coefficient of x

Now, you have a model for x in terms of y, which is what the question is asking for.

x = log(y/4)/log(1.0256)

When y=12, this is .x = log(12/4)/log(1.0256) ≈ 43.46 weeks

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The function will exist on all real numbers. Hence the domain of the exponential function given is all real numbers

The domain are independent values of an expression for which it exist. The standard exponential equation is expressed as y = ab^x

Given the exponential function as shown:

y = 5(3)^x

The function will exist on all real numbers. Hence the domain of the exponential function given is all real numbers

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Given that, Joe travels 68 mi in 4 hours, the number of miles Joe will travel in 45 minutes is 12.75.

Given that;

First, we convert 4 hours to minutes: 4 × 60 = 240 minutes

Now,

Joe travels 68 miles in 240 minutes

Joe will travel x miles in 45 minutes

We have;

x × 240 = 68 × 45

x × 240 = 3060

x = 3060 / 240

x = 12.75 miles.

Given that, Joe travels 68 mi in 4 hours, the number of miles Joe will travel in 45 minutes is 12.75.

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The difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll is 0.375.

In the question, we are given that a person has a 0.125 probability of getting the number 3 on a roll of an eight-sided die.

This means that the theoretical probability of rolling a 3 on an eight-sided die is 0.125.

Also, we are informed that a person rolls the die 24 times, and gets a 3 on one-half of the rolls, that is, 1/2 * 24 = 12 rolls.

Thus, the experimental probability of rolling a 3 on an eight-sided die = 12/24 = 1/2 = 0.5.

In the question, we are asked to find the difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll.

The difference = The experimental probability - The theoretical probability,

or, the difference = 0.5 - 0.125 = 0.375.

Thus, the difference between the theoretical probability and the experimental probability of the person getting the number 3 on a roll is 0.375.

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The operation is 365*24*60*60*1000 = 31536*10^6.

As we know the year has 365 days.

Each day has 24 hours.

Each hour has 60 minutes.

Each minute has 60 seconds.

Each second has 1000 millisecond.

Therefore number of milliseconds in a year is

365*24*60*60*1000 = 31536*10^6.

Hence we get the The operation is 365*24*60*60*1000 = 31536*10^6.

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Step-by-step explanation:

z is the answer of your question 15 is correct answer

He need to charge 24$ per tree if he wants to make a profit of $600.

According to statement

Noah knows that his profit, P, in dollars, are P= −3x^2 + 150 − 1200.

Now, we find the value of x when the profit is $600

Where X represent the Amount he charges per tree.

The given equation is P= −3x^2 + 150 − 1200.

Substitute the value of p is $600 in the above written equation

Then

600 = −3x^2 + 150 − 1200.

After solving the equation by a substitution method the value of X becomes

600+1200-150 = -3(x)^2

x = (1650/3)^1/2

x= 24$

it means the charge per tree is 24$.

So, He need to charge 24$ per tree if he wants to make a profit of $600.

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The differentiation of the stated series expansion of  f term-by-term to obtain the corresponding series expansion for the stated derivative of f is  [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)Xⁿ⁻¹

f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ

⇒ f'(x) = d/dx [[tex]\sum_{n=0}^{\infty}[/tex](-1)ⁿ 3ⁿXⁿ]

= [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿ d/dx (xⁿ)

= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿ(nxⁿ⁻¹)

= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ3ⁿnxⁿ⁻¹

= [tex]\sum_{n=1}^{\infty}[/tex] ((-1)ⁿnxⁿ⁻¹)3ⁿ

= [tex]\sum_{n=1}^{\infty}[/tex] (-1)ⁿ(n3ⁿ)xⁿ-1

In mathematics, differentiation is used to calculate rates of change. In mechanics, for example, velocity is the rate of change of displacement (with regard to time).

The acceleration is the rate of change of velocity (with respect to time).

They are employed in many fields, including

They can be used to represent:

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Full Question:

Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f.

If f(x) = 1/ (1 + 3x) = [tex]\sum_{n=0}^{\infty}[/tex] (-1)ⁿ3ⁿxⁿ