Question

100 PTS PLEASE HELP!!!!

Answer 1

Answer:

Part A) see the explanation

Part B) see the explanation

Step-by-step explanation:

Let

x ----> the distance from one side of the tunnel in feet:

f(x) ---> the height of a tunnel in feet

Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?

1)

we know that

The x-intercepts are the values of x when the value of the function is equal to zero

In the context of the problem, the x-intercepts are the distances from one side of the tunnel when the height of the tunnel is equal to zero

2)

The maximum value of the graph is the vertex

The x-coordinate of the vertex represent the distance from one side of the tunnel when the height is a maximum value

The y-coordinate of the vertex represent the maximum height of the tunnel

In this problem

the vertex is (18,32)

That means

The maximum height of the tunnel is 32 feet and occurs when the distance from one side of the tunnel is 18 feet

3)

we know that

The function is increasing at the interval [0,18)

That means

As the distance from one side of tunnel increases the height of the tunnel increases too

The function is decreasing at the interval (18,36]

That means

As the distance from one side of tunnel increases the height of the tunnel decreases

Part B: What is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent?

we know that

To find the average rate of change, we divide the change in the output value by the change in the input value

the average rate of change using the graph is equal to

$\frac{f(b)-f(a)}{b-a}$

In this problem we have

$a=5$

$b=15$

$f(a)=f(5)=15$  ----> see the attached figure

$f(b)=f(15)=31$  ----> see the attached figure

Substitute

$\frac{31-15}{15-5}=1.6$

That means

The height of the tunnel increases 1.6 feet as the distance from one side of tunnel increases 1 foot in the interval from x=5 to x=15