Step-by-step explanation:
We know that P = 2L + 2W and A = L x W (P = perimeter, A = area, L = length and W = width)
Let x = length and y = width for our problem
P = 2x + 2y
A = xy = 1225
Solving the area function for y gives us y = 1225/x
Plug this into the perimeter function
P = 2x + 2(1225/x)
P = 2x + 2450/x
P = 2x + 2450x-1
Find the derivative of the perimeter function to locate the critical points
P' = 2 - 2450x-2
Now, set this equal to 0.
0 = 2 - 2450x-2
0 = 2 - 2450/x²
2450/x² = 2
Cross multiply to get
2x² = 2450
x² = 1225
x = 35 (we eliminate the -35 because dimensions cannot be negative)
Looking at x < 35, we see that P' is negative. However, when x > 35, P' is positive. This means that x = 35 is an absolute minimum.
When x = 35, y = 1225/x = 1225/35 = 35
The dimensions of the rectangle are 35 x 35 meters
This makes sense because the rectangle with minimum perimeter is a square
The perimeter of this rectangle is 2x + 2y = 2(35) + 2(35) = 70 + 70 = 140 meters.
