Answer:
The maximum area is 800 ft².
Step-by-step explanation:
Lets call L the length of the side parallel to the garage and M the length of the other two sides. The total amount to fence is 2M+L and the area is L*M.
Since Pat got 80ft of fence, then 2M+L = 80, hence L = 80-2M. By replacing the value of L in the formula of the area, we get that
A(M) = M*(80-2M) = -2M² + 80M
The maximum of this function can be obtaining throught derivation, but since it is a quadratic with negative main coefficient, we know that the maximum is the vertex. The x-coordinate of the vertex is '-b/2a' = -80/2*(-2) = 20. The y-coordinate (which represents the maximum area), as a result, is A(20) = -2*20²+80*20 = 800.
800 is the maximum area that can be fenced. Note that M = 20 and L = 80-2L = 40.
