Question

The perimeter of a rectangle is 80 cm. Find the lengths of the sides of the rectangle giving the maximum area.Enter the answers for the lengths of the sides in increasing order.

Answer 1

Answer:

The lengths of the sides are 20 cm and 20 cm

Step-by-step explanation:

Given

Perimeter, P = 80cm

Represent the length and width with L and W, respectively;

$P= 2*(L + B)$

Substitute 80 for P

$80 = 2 * (L + B)$

Divide through by 2

$40 = L + B$

$L + B = 40$

Make L the subject of formula

$L = 40 - B$

Area of a rectangle is calculated as thus;

$Area = L * B$

Substitute 40 - B for L

$Area = (40 - B) * B$

Express this as a function

$A(B) = (40 - B)* B$

$(40 - B)* B = A(B)$

Set A(B) = 0 to determine the roots

Hence;

$(40 - B)* B = 0$

$40 - B = 0$ or $B = 0$

$40 = B$ or $B = 0$

$B = 40$ or $B = 0$

The maximum area of a rectangle occurs at half the sum of the roots;

So;

$B= \frac{B_1 + B_2}{2}$

$B= \frac{40+0}{2}$

$B= \frac{40}{2}$

$B = 20$

Recall that $L = 40 - B$

$L = 40 - 20$

$L = 20$

Hence the lengths of the sides are 20 cm and 20 cm