A rectangle has a perimeter of 40 meters. Find the dimensions that maximize its area.

Step 1: Define the variables.

Let the length be x meters and the width be y meters.

The perimeter is 40 meters, so:
2x + 2y = 40

Divide by 2:
x + y = 20

Solve for y:
y = 20 - x

Step 2: Write the area as a function of one variable.

The area of a rectangle is:
A = xy

Substitute y = 20 - x:
A(x) = x(20 - x)

A(x) = 20x - x^2

Step 3: Differentiate the area function.

A'(x) = 20 - 2x

Step 4: Find the critical point.

Set the derivative equal to zero:
20 - 2x = 0

2x = 20

x = 10

Step 5: Find the corresponding width.

y = 20 - x

y = 20 - 10

y = 10

Step 6: Verify that this gives a maximum.

The second derivative is:
A''(x) = -2

Because A''(x) is negative, the area function is concave down, so the critical point gives a maximum.

Therefore, the rectangle with maximum area has dimensions 10 meters by 10 meters.