Question

Wally’s Burger Shack wants to put up a giant sign by Interstate 81. According to local sign ordinances, any sign visible from the interstate must have a frontal square footage of 529 feet or less. The entire sign will be a gigantic W cut out from billboard material, as shown in the graph that follows, where the top edges of the W are at a height of 55 feet and the boundaries of the W are given by the functions

$$\begin{gathered} f\left(x\right)=0.5{\left(x-12\right)}^2 \\ g\left(x\right)=0.5{\left(x-24\right)}^2 \\ r\left(x\right)={\left(x-12\right)}^2+10 \\ s\left(x\right)={\left(x-24\right)}^2+10 \end{gathered}$$

(a) Write the total area of the front of the W sign in terms of definite integrals. You will need to find the solutions of f (x) = 55, r(x) = 55, s(x) = 55, and g(x) = 55 as part of your work. Be careful about how you divide up the region.

(b) Use your answer to part (a) to calculate the exact frontal square footage of the W sign. Will Wally’s sign meet the local square footage requirements?

Short answer

a

Step-by-step solution

part (a) Step 1. Given Information.

$$\begin{gathered} f\left(x\right)=0.5{\left(x-12\right)}^2 \\ g\left(x\right)=0.5{\left(x-24\right)}^2 \\ r\left(x\right)={\left(x-12\right)}^2+10 \\ s\left(x\right)={\left(x-24\right)}^2+10 \end{gathered}$$

Step 2. part (a).

Substitute $55$for the function value in function $f\left(x\right)$and solve the equation for x.

$$\begin{gathered} f\left(x\right)=0.5{\left(x-12\right)}^2 \\ 55=0.5{\left(x-12\right)}^2 \\ 110={\left(x-12\right)}^2 \\ x=12\pm \sqrt{110} \end{gathered}$$

Step 3. part (b)

a