( Trig. Functions and Angles of Elevation & Depression)
A spotlight (which is mounted at ground level) is pointing to the top of the Empire State Building with an angle of elevation of 78°
. If the horizontal (along the ground) distance between the building and the spotlight is is 309 feet, how tall is the Empire State Building?

The end behaviour of the graph of f(x) \(=\) x⁵ - 8x⁴ \(+\) 16x³  is f(x)→∞ as x→∞; f(x) →-∞ as x→ -∞ , the correct option is (a) .

In the question ,

it is given that ,

the function is f(x) \(=\) x⁵ - 8x⁴ \(+\) 16x³ ,

In the function as x approaches -∞ , x⁵ will also approach -∞ , because negative number raised to the odd power , gives a negative result .

In the function as x approaches ∞ , x⁵ will also approach ∞ , because positive number raised to the odd power , gives a positive result .

that is f(x)→∞ as x→∞; f(x) →-∞ as x→ -∞

let us find the roots of the function ,

f(x) \(=\) 0

x⁵ - 8x⁴ \(+\) 16x³ = 0

taking x³ common ,

x³(x² - 8x \(+\) 16) = 0

x³(x² - 4x - 4x + 16) = 0    .... by using splitting the middle term method

x³(x(x - 4) - 4(x - 4)) = 0

x³(x - 4)(x - 4) = 0

which means x³ \(=\) 0   and x-4 = 0  

that is x=0    and x = 4 .

So , the roots of the function f(x) are x=0 and x=4 .

This means that the graph has a repeated root and so it will touch the x-axis but not at the repeated root of x \(=\) 4.

Therefore , The end behaviour of the graph of f(x) \(=\) x⁵ - 8x⁴ \(+\) 16x³  is f(x)→∞ as x→∞; f(x) →-∞ as x→ -∞ , the correct option is (a) .

The given question is incomplete , the complete question is

What is the end behavior of the graph of f(x) = x⁵ - 8x⁴ + 16x³ ?

(a) f(x)→∞ as x→∞; f(x) →-∞ as x→ -∞

(b) f(x)→∞ as x→-∞0; f(x) → +∞ as x→ +∞

(c) f(x)→∞ as x→ -∞; f(x) →-∞ as x→ +∞

(d) f(x)→-∞ as x→∞ ; f(x)→ -∞ as x→∞

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Answer:

\(\dfrac{5e^2}{2}\)

Step-by-step explanation:

Differentiation is an algebraic process that finds the slope of a curve. At a point, the slope of a curve is the same as the slope of the tangent line to the curve at that point. Therefore, to find the slope of the line tangent to the given function, differentiate the given function.

Given function:

\(y=x^2\ln(2x)\)

Differentiate the given function using the product rule.

\(\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}\)

\(\textsf{Let\;$u=x^2}\)\(\textsf{Let\;$u=x^2$}\implies \dfrac{\text{d}u}{\text{d}x}=2x\)

\(\textsf{Let\;$v=\ln(2x)$}\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{2}{2x}=\dfrac{1}{x}\)

Input the values into the product rule to differentiate the function:

\(\begin{aligned}\dfrac{\text{d}y}{\text{d}x}&=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}\\\\&=x^2 \cdot \dfrac{1}{x}+\ln(2x) \cdot 2x\\\\&=x+2x\ln(2x)\end{aligned}\)

To find the slope of the tangent line at x = e²/2, substitute x = e²/2 into the differentiated function:

\(\begin{aligned}x=\dfrac{e^2}{2}\implies \dfrac{\text{d}y}{\text{d}x}&=\dfrac{e^2}{2}+2\left(\dfrac{e^2}{2}\right)\ln\left(2 \cdot \dfrac{e^2}{2}\right)\\\\&=\dfrac{e^2}{2}+e^2\ln\left(e^2\right)\\\\&=\dfrac{e^2}{2}+2e^2\\\\&=\dfrac{5e^2}{2}\end{aligned}\)

Therefore, the slope of the line tangent to the graph of y = x²ln(2x) at the point where x = e²/2 is:

\(\boxed{\dfrac{5e^2}{2}}\)

The number of square meters that the space that Araceli rents is, is 30. 66 square meters.

The space that Araceli rents has a rectangular space so to find the area of the rectangular space that Araceli rents, we need to multiply the length by the width.

The given width is 4. 2 meters and the length is 7. 3 meters.

The area of the space in square meters is therefore :

Area = Length × Width

Area = 7. 3 meters × 4. 2 meters

Area = 30. 66 square meters

In conclusion, the space that Araceli rents has an area of 30. 66 square meters.

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Answer:

Step-by-step explanation:

I and II

Pages in a book and leaves on a tree can be counted precisely. They are discrete.

The height of a person cannot be measured exactly. We can get very good

approximations (to a lot of decimal places) but never exact. This is continuous.

The biker needs to travel 45 km more to complete his journey.

Given that, a biker traveled 2/5 of the road on the first day, then traveled 5 kilometers less on the next day, which is equal to 3/8 of the road, we need to find how much he need to cover more,

Let the total distance of the road be x,
So,

2x/5 - 5 = 3x/8

2x/5 - 3x/8 = 5

Solving for x,

Multiply by 40 to both sides,

16x-15x = 200

x = 200

Therefore, the total distance of the road is 200 km.

Since, he travelled =

200 (2/5) = 80 km + 200 (3/8) = 75 km = 155 km

Therefore, he needs to travel 200-155 = 45 km more.

Hence, the biker needs to travel 45 km more to complete his journey.

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TJohn's age is approximately 23.33 years, and Sharon's age is approximately 93.33 years.

And the correct graph is D.

To represent the given problem as a system of equations, we can use the following information:

John is 70 years younger than Sharon: j = s - 70

Sharon is 4 times as old as John: s = 4j

Let's plot the graph for this system of equations:

First, let's solve equation (2) for s:

s = 4j

Now substitute this value of s in equation (1):

j = s - 70

j = 4j - 70

3j = 70

j = 70/3

Substitute the value of j back into equation (2) to find s:

s = 4j

s = 4(70/3)

s = 280/3

The solution to the system of equations is j = 70/3 and s = 280/3

In the graph d, the solution to the system of equations is represented by the point (70/3, 280/3), which is approximately (23.33, 93.33) on the graph.

Therefore, John's age is approximately 23.33 years, and Sharon's age is approximately 93.33 years.

And the correct graph is D.

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The 90% confidence interval for the true proportion of all adults in the town who have health insurance is approximately 0.7294 to 0.8804.

Using the formula for calculating a confidence interval for a proportion.

The formula for the confidence interval is:

Confidence Interval = Sample proportion ± Margin of error

where:

Sample proportion = Number of adults with health insurance / Total sample size

Margin of error = Critical value * Standard error

The critical value depends on the desired confidence level. For a 90% confidence level, the critical value can be obtained from the standard normal distribution table, which corresponds to a Z-value of 1.645.

The standard error can be calculated using the formula:

Standard error =\(\sqrt{} ((p * (1 - p)) / n)\)

Given:

Number of adults with health insurance (sample proportion) = 66

Total sample size = 82

Confidence level = 90%

Now, let's calculate the confidence interval:

Sample proportion = 66 / 82 ≈ 0.8049

Standard error = \(\sqrt{}\)((0.8049 * (1 - 0.8049)) / 82) ≈ 0.0459

Margin of error = 1.645 * 0.0459 ≈ 0.0755

Lower bound of the confidence interval = Sample proportion - Margin of error ≈ 0.8049 - 0.0755 ≈ 0.7294

Upper bound of the confidence interval = Sample proportion + Margin of error ≈ 0.8049 + 0.0755 ≈ 0.8804

Therefore, the 90% confidence interval for the true proportion of all adults in the town who have health insurance is approximately 0.7294 to 0.8804.

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Answer: Approximately 25.1 m^3

Step-by-step explanation:

The volume of a cone is \(V=\pi r^2\frac{h}{3}\). We know the height, but we don't know the radius. Fortunately, we are given the circumference of the base circle.

\(C=2\pi r\)

\(9=2\pi r\)

\(r=\frac{9}{2\pi}\)

Now that we have the r, we can plug it into the volume equation.

\(V=\pi r^2\frac{h}{3}\)

\(V=\pi (\frac{9}{2\pi } )^2 (\frac{11.7}{3} )\)

After you plug this into the calculator, you get approximately 25.1 m^3.

Answer:

A-10.6 Kilometers

Step-by-step explanation:

1 kilometer =39370.079 inches

Answer:

Step-by-step explanation:

y = -1x + 1

y = -x + 1

I'm sure  :P

Answer:

y=-1x+1

Step-by-step explanation:

the slope is 1/1 and line is declining hence the negative

Answer:

3/4

Step-by-step explanation:

The gradient of  y = 16 - 8x is - 8

The gradient of y - 5 = 6x is 6

y = mx + c form of y - 5 = 6x is

The slope is another name for the gradient. Any straight line's gradient illustrates or demonstrates how steep it is. The gradient is said to be bigger if any line is steeper.

In other words, the ratio of vertical change to horizontal change is known as Gradient.

The gradient of line y = mx + c is 'm'  

Here we have

Equation (1) y = 16 - 8x

Compare the given equation with y = mx + c

=> Gradient / Slope (m) = - 8

Equation (2) y = 7x + 8

Compare the given equation with y = mx + c

=> Gradient/ Slope (m) = 7

Equation (3) y - 5 = 6x

Convert the given equation in the form of y = mx + c

=> y - 5 = 6x  

Add 5 on both sides

=> y = 6x + 5

When we compare the given equation with y = mx + c

=> Gradient / Slope (m) = 6

Therefore,

The gradient of  y = 16 - 8x is - 8

The gradient of y - 5 = 6x is 6

y = mx + c form of y - 5 = 6x is y = 6x + 5.  

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Answer:

the type of growth defined by the equation is Exponential growth.

Step-by-step explanation:

Exponential growth :-Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself

expression for exponential growth -

f(x) = k(1+t)^x

f(x) = exponential growth

k = initial amount

t = rate of growth

x = no  of time interval

therefor,

the equation shown in the question represents exponential growth with increasing time.

hence, Exponential growth is the correct answer.

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Answer:

a complementary good

Step-by-step explanation:

Howard spends $70,000 per year on 5 suits. When his income reaches $200,000, he purchases 15 suits per year. Suits are classified as a normal good based on this information.

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Answer:

(d) -12

Step-by-step explanation:

3a = _____

a = -4

3(-4) = ___

3(-4) = -12

The answer is (d) -12

Answer:

120 balls

Step-by-step explanation:

Ten dozen is 120.

Answer:

green line or "b"

Step-by-step explanation:

we can set this up into the slope intercept form

y=mx+b

where m is the slope and b is y intercept

12=3x+4y

-3x -3x

-3x+12=4y

/4.  /4.  /4

-3/4x+3=y

we can see the only line with a negative slope and a y intercept at 3 is the green line or "b"

hopes this helps please mark rainliest

Answer:

The distances AB and BC are

√

52

and therefore isosceles.

Explanation:

If we label them A(13,-2) B(9-8) and C(5-2)

The distance between A and B:13⇒9=4−2⇒−8=42+62=52 so the distance is √52

The distance between B and c:9⇒5=4−8⇒−2=642+62=52

the distance is

√

52

The distance between A and C is 8 as they are both on -2 for

y

The distances AB and BC are

√

52

and therefore isosceles.Step-by-step explanation:

Answer:

0.125 cubic feet

Step-by-step explanation:

Since the container has a shape of a cube with a measure of its side 0.5 ft, then;

Volume of the container, V = \(s^{2}\)

where s is the measure of its side.

s = 0.5 ft, then;

V = \((0.5)^{3}\)

    = 0.125

V = 0.125 cubic feet

The cubic feet of slime that would fit in the container is 0.125.

1. The quadratic function for the Area in terms of x: A(x) = 24x.

2. The cross-sectional area is maximized when the depth of the gutter is 0.

3. The maximum cross-sectional area is square inches 0.

To determine the depth of the gutter that maximizes its cross-sectional area and allows the greatest amount of water to flow, we need to follow a step-by-step process.

1. Write a quadratic function for the area in terms of x:

The cross-sectional area of the gutter can be represented as a rectangle with a width of 24 inches and a depth of x. Therefore, the area, A(x), is given by A(x) = 24x.

2. The cross-sectional area is maximized when the depth of the gutter is:

To find the value of x that maximizes the area, we need to find the vertex of the quadratic function. The vertex of a quadratic function in form f(x) = ax² + bx + c is given by x = -b/(2a). In our case, a = 0 (since there is no x² term), b = 24, and c = 0. Thus, the depth of the gutter that maximizes the area is x = -24/(2 * 0) = 0.

3. The maximum cross-sectional area is square inches:

Substituting the value of x = 0 into the quadratic function A(x) = 24x, we get A(0) = 24 * 0 = 0. Therefore, the maximum cross-sectional area is 0 square inches.

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Answer:

the market value of all final goods and services produced within the United States in a given period of time.

The quadratic equation is given as

Using the square roots property, we need to find the square root of both sides.

Therefore, we get

Hence, we can solve as

The roots can be gotten as

or

The answer is

Answer:

64 inches

Step-by-step explanation:

There are 12 inches in a foot

12 x 5 is 60

64 is greater then 60

The fuction f(g(x)) is given by

The function g(f(x)) is

The functions f and g will be inverses of each other if the function values obtained is x in both cases. But, we got 4x+30 for f(g(x) and 4x-2 for g(f(x)), which are not equal to x. So, f and g are not inverses of each other.

The important variables are the two types of test questions which can be represented as :

Variables are used to represent unknown values which could be worked out in a mathematical expression or problem.

The variables or unknown in this case are the type of test questions. which are : m for multiple choice, s for short answer

Therefore, the correct option is C. m for multiple choice, s for short answer

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