Question

A road is made in such a way that the center of the road is higher off the ground than the sides of the road, in order to allow rainwater to drain. A cross-section of the road can be represented on a graph using the function f(x) = (x – 16)(x + 16), where x represents the distance from the center of the road, in feet. Rounded to the nearest tenth, what is the maximum height of the road, in feet?

Answer 1

Answer:

Step-by-step explanation:

Given that a road is made in such a way that the center of the road is higher off the ground than the sides of the road, in order to allow rainwater to drain. A cross-section of the road can be represented on a graph using the function f(x) = (x – 16)(x + 16), where x represents the distance from the center of the road, in feet.

The height of the road i.e. f(x) is dependent on x, the distance from centre

To find the maximum height of road, let us use derivative tes

f'(x) = (x-16)1+(x+16)  (using product rule)

      =2x

f"(x) =2>0

Since second derivative is positive, we have a minimum at x=0

There is no local maximum for the road

But x cannot take values less than 16.  Hence x lies after 16 only.

The value of funciton f(x) = (x – 16)(x + 16)

can have minimum when x=16 and maximum when x =infinity

Answer 2

F(x) = (x-16)(x+16)
f(x) = (x - 16)²

To find the maximum, we set the first derivative equal to 0 and then calculate x.
f'(x) = 2(x - 16)
0 = 2(x - 16)
x = 16

The maximum height of the road is 16 feet.