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.
