Chapter 9.8, Problem 3GP

To determine

To calculate: To graph the equation that models the area of the framed picture and then find the area of the picture if the width of picture is $6$ inches.

Answer to Problem 3GP

Equation that models the area of the framed picture is $y=\frac{2}{3}{x}^2$

Area of picture frame when width is $6$ inches is $54$ square inches.

Explanation of Solution

Given information: Mei is building a picture frame, and she wants the width to be $\frac{2}{3}$ the length

Formula Used:

Area of a rectangle is equal to product of length and width

Calculation:

Let the length of picture frame be $x$

Given that width of picture frame is $\frac{2}{3}$ the length

Thus, $width=\frac{2}{3}x$

Picture frame is in a shape of rectangle.

Let the are of picture frame be $y$

Thus, equation that models the area of the framed picture is given as

  $y=x\times \frac{2}{3}x$

On solving the above equation,

  $y=\frac{2}{3}{x}^2$

In order to find the area of picture frame when width of picture frame is $6$ inches.

Thus, length is calculated as

  $Width=\frac{2}{3}x$

Substituting the values to calculate the length as below:

  $\begin{array}{l}6=\frac{2}{3}x\\ x=6\times \frac{3}{2}\\ x=9\end{array}$

Thus, area when width is $6$ inches is

  $\begin{array}{l}Area=9\times 6\\ Area=54\end{array}$

Conclusion:

Hence, equation that models the area of the framed picture is $y=\frac{2}{3}{x}^2$

Area of picture frame when width is $6$ inches is $54$ square inches.