To find the maximum area of a rectangle with a given perimeter, we use the fact that for a fixed perimeter, the maximum area is achieved when the rectangle is a square.
Step 1: Use the formula for the perimeter of a rectangle:
$P = 2(l + b)$
$620 = 2(l + b)$
$l + b = 310$
Step 2: Express the area in terms of one variable:
The area of the rectangle is given by:
$A = l * b$
$l = 310 - l $
$A = l * (310 - l) = 310l - l^2$
Step 3: Maximize the area:
To maximize the area, we take the derivative of with respect to and set it equal to zero:
$\frac{dA}{dl} = 310 - 2L = 0$
$2l = 310 \implies l = 155$
Step 4: Calculate the maximum area:
The rectangle is a square with side length mm. The area is:
$A = 155 * 155 = 24,025 \text{ mm}^2$
