Answer:
For width W and length L, we have:
W = L = √(75ft^2) = 8.66ft
Step-by-step explanation:
For a rectangle of length L and width W, the area is:
A = L*W
and the perimeter is:
P = 2*L + 2*W
First, we know that the area of our rectangle is:
A = 75ft^2 = L*W
And we want to minimize the perimeter of our rectangle, then we need to minimize:
P = 2*L + 2*W
From the equation:
75ft^2 = L*W
We can isolate one of the variables, let's isolate L
L = (75ft^2)/W
We could replace this in the perimeter equation:
P(W) = 2*( (75ft^2)/W) + 2*W
P(W) = (150 ft^2)/W + 2*W
To find the minimum of the perimeter we need to look at the zero of the first derivative of P(W).
P'(W) = dP(W)/dW = -(150ft^2)/W^2 + 2
Now we need to find the value of W such that:
P'(W) = 0
0 = -(150ft^2)/W^2 + 2
(150 ft^2)/W^2 = 2
(150ft^2) = 2*W^2
(150ft^2)/2 = W^2
75 ft^2 = W^2
√(75ft^2) = W = 8.66ft
And remember that:
L = (75ft^2)/W
replacing with W = √(75ft^2)
L = (75ft^2)/√(75ft^2) = √(75ft^2)
Then:
W = L = √(75ft^2) = 8.66ft
