Jason is making a model of a bridge as shown below.

The function [tex]$h$[/tex] represents the height of the bridge, in feet, and [tex]$x$[/tex] represents the distance along the base of the model from the leftmost edge, in feet.
[tex]\[ h(x) = -0.5(x - 4)^2 + 2 \][/tex]

Which of the following statements is true about this situation?

A. The bridge touches the base of the model at 2 feet and 4 feet from the leftmost edge of the model.

B. The bridge touches the base of the model at 0.5 feet and 4 feet from the leftmost edge of the model.

C. The bridge touches the base of the model at 2 feet and 6 feet from the leftmost edge of the model.

D. The bridge touches the base of the model at 0.5 feet and 6 feet from the leftmost edge of the model.



Answer :

To determine the points where the bridge touches the base of the model, we need to find the values of [tex]\( x \)[/tex] where the height [tex]\( h(x) \)[/tex] equals zero.

Given the function:

[tex]\[ h(x) = -0.5(x-4)^2 + 2 \][/tex]

we set [tex]\( h(x) = 0 \)[/tex] to find these points:

[tex]\[ 0 = -0.5(x-4)^2 + 2 \][/tex]

First, move 2 to the left side of the equation:

[tex]\[ -2 = -0.5(x-4)^2 \][/tex]

Next, divide both sides by -0.5:

[tex]\[ 4 = (x-4)^2 \][/tex]

Now, take the square root of both sides to solve for [tex]\( x \)[/tex]:

[tex]\[ \sqrt{4} = x - 4 \][/tex]

This gives us two possible solutions:

[tex]\[ x - 4 = 2 \quad \text{or} \quad x - 4 = -2 \][/tex]

Solving for [tex]\( x \)[/tex]:

[tex]\[ x = 4 + 2 = 6 \][/tex]

[tex]\[ x = 4 - 2 = 2 \][/tex]

Therefore, the bridge touches the base of the model at [tex]\( x = 2 \)[/tex] feet and [tex]\( x = 6 \)[/tex] feet from the leftmost edge of the model.

So, the correct statement is:

C. The bridge touches the base of the model at 2 feet and 6 feet from the leftmost edge of the model.