Question

a) Determine the dimensions and area of the rectangle with a given perimeter of 24 cm and a maximum area.b) Determine the dimensions of a rectangle with a minimum perimeter and a given area of 110.25 m2.

Answer 1

b. The area of the rectangle is 110.25 m²

Consider the figure shown below

The Perimeter of the rectangle will be

`\begin{gathered} P=2(x+(110.25)/(x)) \\ P=2x+(220.5)/(x) \end{gathered}`

Differentiate with respect to x

`(dP)/(dx)=2-(220.5)/(x^2)`

Equate dP/dx to zero

It follows

`2-(220.5)/(x^2)=0`

Solve the equation for x

`\begin{gathered} 2=(220.5)/(x^2) \\ 2x^2=220.5 \\ x^2=110.25 \\ x=√(110.25) \\ x=10.5 \end{gathered}`

Therefore, the value of x=10.5 m

The length of the rectangle is 10.3 m

The width of the rectangle is

`(110.25)/(10.5)=10.5`

Hence the width of the rectangle is 10.5 m

Therefore the dimension of the rectangle is 10.5 m by 10.5 m