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

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