Question

The area of a rectangle is 81cm^2 what are the dimensions if the perimeter is minimum. What is the value of these minimum perimeter?

Answer 1

The perimeter is 36 cm

Explanation:

Given;

area of the rectangle, A = 81 cm²

let the Length, = L

let the breadth = b

Perimeter, P = 2L + 2b

Area, A = Lb = 81

`L = (81)/(b) \\\\P = 2((81)/(b) ) \ + \ 2b\\\\P = (162)/(b) \ + \ 2b\\\\P = 162(b^(-1)) \ + \ 2b\\\\minimize \ the \ perimeter \ to \ obtain \ the \ critical \ points;\\\\P' = -162 (b^(-2)) \ + \ 2\\\\P' = (-162)/(b^2) \ + \ 2\\\\P' = (-162 \ + \ 2b^2)/(b^2) \\\\at \ critical \ points; P' = 0\\\\(-162 \ + \ 2b^2)/(b^2) = 0\\\\-162 \ + \ 2b^2 = 0\\\\2b^2 = 162\\\\b^2 = 81\\\\b = +/- \ √(81) \\\\b = +/- \ \ 9\\\\`

`since \ the \ breadth \ will \ be \ positive \ ; b = 9 \ cm`

L = 81/9

L = 9 cm

The perimeter = 2 (L + b)

= 2 (9 + 9)

= 2(18)

= 36 cm