The area A, in square meters, of a rectangle with a perimeter of 160 meters is given by the equation A = 80w - w2, where w is the width of the rectangle in meters. What is the width of a rectangle if its area is 700 m2?

the width is 10 m

Step-by-step explanation:

if the relationship between area and width is

A = 80*w - w²

for an area A=700 m², we have

700 m² = 80*w - w²

w² - 80*w + 700 m² = 0

aw² + b*w + c = 0

where a=1, b=-80 and c=700

this quadratic equation has as solution the following formula

w = [-b ± √ (b² - 4*a*c) ] / (2*a)

replacing values

w = [80 ± √ (80² - 4*1*700) ] / (2*1) = (80 ± 60) / 2

then

w₁ = (80 - 60) / 2 = 10 m

w₂ = (80 + 60) / 2 = 70 m

since the area has the form A = length * width = 80*w - w² = (80 - w) * w

then the length of the rectangle is

length = 80 - w

for w₁=10 m → length = 80 - 10 = 70 m

for w₁=70 m → length = 80 - 70 = 10 m

by definition the shorter side is the width (and the longer one, the length), therefore the only possible option is the first one.

Thus the width is 10 m