Answer: A (max) = 1/2 ( 8 * 8 ) = 32 ft²
Step-by-step explanation:
We have two constrains
a) The height of the sail ( support) cannot exceed 12 feet
b) The total weight of the supports cannot exceed 80 pounds
The weight of a foot of aluminum is 5 .
If we call:
x lenght of base support, y height of the vertical support and the sail shape (is a triangle) we have
A = 1/2* ( x + y )
We know that (given a constant perimeter rectangle, the maximun area is the square
A = x * y P = 2x +2y y = (P - 2x ) ÷ 2
A = x * ( P - 2x ) ÷ 2 ⇒ A = (Px -2x²) ÷ 2
if we get derivative A´(x) = (P-4x)/2 ⇒ (P-4x)/2 = 0
P = 4x and x = P/4
Now we can look an square as two straight triangles joined by the diagonal and these two triangles are of area maximun.
Therefore in our case the sail must be of 8 x 8 feet (base and height)
A (max) = 1/2 ( 8 * 8 ) = 32 ft² and we will keep the condition of weigh
8 + 8 = 16 feet and the supports weights 16 * 5 = 80 pounds
