Chapter 4.3, Problem 67PPS

To determine

To Find: The dimension of the panting with the border included.

Answer to Problem 67PPS

The length of the border is $20\text{in}$ and the breadth of the border is $15\text{in}$ .

Explanation of Solution

Given:

The border is distributed evenly and has the same area as the panting.

Calculation:

Consider the length of the panting is 15 inch and the breadth is $10\text{in}$ .

Then, the area of the panting is,

$\begin{array}{c}A=\left(10\right)\left(15\right)\\ =150\end{array}$

Consider the width of the border is x then the width is,

$A=\left(10-2x\right)\left(15-2x\right)-150$

Since, the area is equal then,

$\begin{array}{l}150=\left(10-2x\right)\left(15-2x\right)-150\\ 4x^2+50x-150=0\\ 2x^2+25x-75=0\\ \left(2x-5\right)\left(x+15\right)=0\end{array}$

Solve further as,

$\begin{array}{c}x=\frac{5}{2}\\ =2.5\end{array}$

Then,

$\begin{array}{l}x+15=0\\ x=-15\end{array}$

Thus, the length of the border is $\left(15+2\left(2.5\right)\right)=20\text{in}$ and the breadth of the border is $\left(10+2\left(2.5\right)\right)=15\text{in}$ .