Chapter 9.6, Problem 6STP

For Exercises 1-6, choose the correct letter.

The perimeter of a rectangle is 54 cm. The area of the same rectangle is 176 $c{m}^2$.

What are the dimensions of the rectangle?

F. 11 cm by 16 cm

H. 5.5 cm by 32 cm

G. 8 cm by 22 cm

I. 4 cm by 44 cm

To determine

To find : The dimensions of the rectangle.

Answer to Problem 6STP

The correct option is (F).

Explanation of Solution

Given information :

The perimeter of a rectangle is 54 cm. The area of the same rectangle is 176 cm².

Calculation :

Let the length and breadth of the rectangle be l and b respectively.

Since, the perimeter is 54. Therefore,

  $\begin{array}{l}2l+2b=54\\ 2\left(l+b\right)=54\\ l+b=\frac{54}{2}\\ l+b=27\\ b=27-l\mathrm{......}\text{ (1)}\end{array}$

Also, area is 176 cm². Therefore,

  $\begin{array}{l}l\times b=176\\ l\times \left(27-l\right)=176\text{ (From 1)}\\ 27l-l^2=176\\ 27l-l^2-176=0\\ l^2-27l+176=0\\ l^2-11l-16l+176=0\\ l\left(l-11\right)-16\left(l-11\right)=0\\ \left(l-11\right)\left(l-16\right)=0\\ l-11=0\text{ and }l-16=0\\ l=11\text{ and }l=16\end{array}$

From (1) the corresponding values of breadth is

For $l=11$

  $\begin{array}{l}b=27-11\\ b=16\end{array}$

For $l=16$

  $\begin{array}{l}b=27-16\\ b=11\end{array}$

Therefore, the dimension of the rectangle are 11 cm by 16 cm.

Hence,

The correct option is (F).