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
, where 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?
0.1
0.8
1.3
1.6

Ask by Bowman Weston. in the United States

Mar 28,2025

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
, where 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?
0.1
0.8
1.3
1.6

Ask by Bowman Weston. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

We are given the function  
$$
f(x) = -\frac{1}{200}(x-16)(x+16)
$$  
This function represents a downward-opening parabola, so its maximum occurs at the vertex.

1. First, express the function in standard quadratic form. Expand the product:
   $$
   (x-16)(x+16) = x^2 - 256.
   $$
   Therefore,
   $$
   f(x) = -\frac{1}{200}(x^2 - 256) = -\frac{1}{200}x^2 + \frac{256}{200}.
   $$

2. Simplify the constant term:
   $$
   \frac{256}{200} = 1.28.
   $$
   So, the function becomes:
   $$
   f(x) = -\frac{1}{200}x^2 + 1.28.
   $$

3. Since the coefficient of $$ x^2 $$ is negative, the maximum value occurs at $$ x=0 $$:
   $$
   f(0) = -\frac{1}{200}(0)^2 + 1.28 = 1.28.
   $$

4. Rounding $$ 1.28 $$ to the nearest tenth gives:
   $$
   1.3.
   $$

Thus, the maximum height of the road, in feet, is $$1.3$$.
The maximum height of the road is 1.3 feet.

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