What is the length and width of a rectangle that has a perimeter of 20 inches and an area of 24.4524 square inches showing work with final answers?

Answer 1

What do we know about the perimeter of a rectangle?

perimeter = 2 × (length + width)

→ 2 × (length + width) = 20 in

→ length + width = 10 in

→ length = 10 in - width

What do we know about the area of a rectangle:

area = length × width

→ length × width = 24.4524 in²

But from the perimeter we know the length in terms of the width and can substitute it in:

→ (10 in - width) × width = 24.4524 in²

→ 10 in × width - width² = 24.4524 in²

→ width² - 10 in × width + 24.4524 in² = 0

This is a quadratic which can be solved by using the formula:

ax² + bx + c → x = (-b ±√(b² - 4ac)) / (2a)

→ width = (-10 ±√(10² - 4 × 1 × 24.4524)) / (2 × 1) in

→ width = -5 ± ½√(100 - 97.8096) in

→ width = -5 ±½√2.1904 in

→ width = -5 ± 0.74 in

→ width = 4.26 in or 5.74 in

→ length = 10 in - 4.26 in = 5.74 in or 10 in - 5.74 in = 4.26 in (respectively)

By convention the width is the shorter length (though it doesn't have to be) making the width 4.26 in and the length 5.74 in.

Thus the rectangle is 5.74 in by 4.26 in