Question

Find the dimensions of a rectangle with perimeter 108 m whose area is as large as possible. (if both values are the same number, enter it into both blanks.)

Answer 1

The dimensions of a rectangle with perimeter 108 mare $\boxed{27{\text{ m}}}$ and $\boxed{27{\text{ m}}}.$

Further explanation:

The perimeter of the rectangle can be expressed as follows,

$\boxed{{\text{Perimeter}} = 2 \times \left( {x + y} \right)}$

Given:

The perimeter of the rectangular is $108{\text{ m}}.$

Explanation:

Consider the length of the rectangular field be “x”.

Consider the width of the box as “y”.

The perimeter of the rectangular is 108 m.

$\begin{aligned}{\text{Perimeter}} &= 2 \times \left( {x + y} \right)\\108 &= 2x + 2y\\108 - 2y &= 2x\\\frac{{108 - 2y}}{2}&= x \\ 54 - y&= x\\\end{aligned}$

The area of the rectangular field can be expressed as follows,

$\begin{aligned}{\text{Area}} &= \left( x \right) \times \left( y \right)\\&= \left( y \right) \times \left( {54 - y} \right)\\&= 54y - {y^2}\\\end{aligned}$

Differentiate the above equation with respect to y.

$\begin{aligned}\frac{d}{{dy}}\left( {{\text{Area}}} \right)&=\frac{d}{{dy}}\left( {54y - {y^2}} \right)\\&= 54 - 2y\\\end{aligned}$

Again differentiate with respect to y.

$\begin{aligned}\frac{{{d^2}}}{{d{y^2}}}\left( {Area} \right) &= \frac{{{d^2}}}{{d{y^2}}}\left( {54 - 2y}\right)\\&= - 2\\\end{aligned}$

Substitute the first derivative equal to zero.

$\begin{aligned}\frac{d}{{dy}}\left({{\text{Area}}} \right)&= 0\\54 - 2y&= 0\\54&= 2y\\\frac{{54}}{2}&= y\\27&= y\\\end{aligned}$

The value of x can be calculated as follows,

$\begin{aligned}x&= 54 - 27\\&= 27\\\end{aligned}$

The dimensions of a rectangle with perimeter 108 mare $\boxed{27{\text{ m}}}$ and $\boxed{27{\text{ m}}}.$

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Application of Derivatives

Keywords: square, area, rectangle, large as possible, same number, one side, shortest length, perimeter, circumference.

Answer 2

The dimensions of the rectangle with perimeter 108 and whose area is large as possible if the both values are the same number. For the rectangle to have maximum area, the dimensions should be equal.
Let the dimensions be x
thus the perimeter will be:
2(x+x)=108
2(2x)=108
4x=108
x=27
thus for the rectangle to have a maximum area the dimensions should be 27m by 27 m