How to Find Length and Width of a Rectangle by its Area and Perimeter

QUESTION: A postcard has an area of 24 square inches. Its perimeter is 20 inches. What are the dimensions of the postcard inches by inches?

SOLUTION:

Let $L$ is the length of the postcard

And $W$ is the width of the postcard

Then

Area of a postcard = $L$ × $W$ = 24 in² ———————(1)

Perimeter of a postcard = $2L$ + $2W$ = 20 in ————————-(2)

From Equation (1)

$L$ = $\frac{24}{W}$ ————————(3)

Put value of L from equation (3) in equation (2)

$2L$ + $2W$ = 20

$2\frac{24}W$ + $2W$ = 20

$\frac{24}W$ + $W$ = 10

24 + $W^2$ = $10W$

$W^2$ – $10W$ + 24 = 0

$W^2$ – $6W$ – $4W$ + 24 = 0

$W$ ( $W$ – 6) -( $4W$ – 6) = 0

( $W$ – 4) ( $W$ – 6) = 0

Thus either

$W$ – 4 = 0 which implies $W$ = 4

$W$ – 6 = 0 which implies $W$ = 6

If $W$ = 4

Put this value of $W$ in equation (3)

$L$ = $\frac{24}W$ ⇒ $\frac{24}4$ = 6 in

If $W$ = 6

Put this value of $W$ in equation (3)

$L$ = $\frac{24}W$ ⇒ $\frac{24}6$ = 4 in

Thus if a postcard has an area of 24 square inches and perimeter of 20 inches, the dimensions of the postcard will be 6 inches by 4 inches.

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