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
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
In this problem we have
[tex]a=5[/tex]
[tex]b=15[/tex]
[tex]f(a)=f(5)=15[/tex] ----> see the attached figure
[tex]f(b)=f(15)=31[/tex] ----> see the attached figure
Substitute
[tex]\frac{31-15}{15-5}=1.6[/tex]
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
