Chapter 14, Problem 28PT

To determine

To calculate: The expression for the area of the garden after increase each side of the garden by $5\text{feet}$ .

Answer to Problem 28PT

The expression for new area of the garden is $6x^2+25x+25$ .

Explanation of Solution

Given Information:

The ratio length to the width of the garden $=3:2$

Formula Used:

  $\text{Area}\text{of}\text{a}\text{rectangle}\left(A\right)=L\times W$ , where $L$ is the length and $W$ is the width of the rectangle.

Distribution formula is:

  $\begin{array}{c}\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)\\ =a\times c+a\times d+b\times c+b\times d\end{array}$

Calculation:

The given ratio length to the width of the garden $=3:2$ .

Let $x$ be the unit length of the garden.

Thus, the length of the garden is $3x$ and width is $2x$ .

Now, it is given that the length and width is increased by $5\text{feet}$ .

Thus, the new length and widthof the garden are $3x+5\text{and}2x+5$ respectively.

Now,find the area of the garden. Use the formula $A=L\times W$ .

  $\text{Area}\text{of}\text{the}\text{garden}=\left(3x+5\right)\left(2x+5\right)$

Find the simplified form of the expression $\left(3x+5\right)\left(2x+5\right)$ .

Use the distribution formula $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)=a\times c+a\times d+b\times c+b\times d$ .

  $\begin{array}{c}\left(3x+5\right)\left(2x+5\right)=3x\left(2x+5\right)+5\left(2x+5\right)\\ =\left(3x\right)\left(2x\right)+\left(3x\right)5+5\left(2x\right)+5\times 5\\ =6x^2+15x+10x+25\\ =6x^2+25x+25\end{array}$

Hence, the required expression for newarea of the gardenis $6x^2+25x+25$ .