Question

Consider all rectangles with an area of 256 cm squared. let x be the length of one side of such a rectangle. express the perimeter as a function of x and determine the dimensions of the rectangle that has the least perimeter

Answer 1

Answer:

- P = 2x + 512/x

- 16cm by 16cm

Step-by-step explanation:

The formula for calculating the area of a rectangle = Length * width

Area = LW

256 = xy .... 1

y = 256/x

x is the length

y is the width

Perimeter of the rectangle = 2(x+y)

P = 2x + 2y

P = 2x + 2(256/x)

P = 2x + 512/x

Hence the perimeter as a function of x is P = 2x + 512/x

For the rectangle to have a least perimeter, this means dP/dx = 0

dP/dx = 2 - 512/x²

0 = 2 -512/x²

2 = 512/x²

2x² = 512

x² = 256

x = √256

x = 16

Since xy = 256

y = 256/16

y = 16

Hence the dimensions of the rectangle that has the least perimeter is 16cm by 16 cm