Q. 79 - Precalculus Enhanced with Graphing Utilities

Question

The perimeter of a rectangle is $16$inches, and its area is $15 inche{s}^2$. What are its dimensions?

Step-by-Step Solution

Answer

The dimensions are $5inches by 3 inches$.

Step 1. Given information.

Assume the length and breadth of the rectangle is l and b.

From the question,

$$\begin{gathered} 2\left(l+b\right)=16 ...... \left(i\right) \\ l\times b=15 ...... \left(ii\right) \end{gathered}$$

Solve equation (ii) for l,

$l=\frac{15}{b}$

Substitute the value of l in equation (i),

$$\begin{gathered} 2\left(\frac{15}{b}+b\right)=16 \\ \frac{15+b^2}{b}=8 \\ 15+b^2-8b=0 ...... \left(iii\right) \end{gathered}$$

Step 2. Factor equation (iii).

On factoring equation (iii), we get,

$$\begin{gathered} \left(b-3\right)\left(b-5\right)=0 \\ b=3,5 \end{gathered}$$

Substitute $b=3$ in equation (ii),

$$\begin{gathered} l\times \left(3\right)=15 \\ l=5 \end{gathered}$$

Substitute $b=5$ in equation (ii),

$$\begin{gathered} l\times \left(5\right)=15 \\ l=3 \end{gathered}$$

Therefore, the dimension are $5inches by 3 inches$.