Chapter 9.8, Problem 10P

To determine

To calculate:

The dimensions of original piece of cardboard

Answer to Problem 10P

The dimensions of original piece of cardboard are $20$ and $28$ .

Explanation of Solution

Given information:

  $\begin{array}{l}A=560c{m}^2\\ V=960c{m}^3\end{array}$

Formula used:

The area of rectangle is given by the following expression:

  $A=xy$

Where,

Length of rectangle: $x$

Width of rectangle: $y$

The volume of rectangle is given by the following expression:

  $V=xyh$

Where,

Length of rectangle: $x$

Width of rectangle: $y$

Height of rectangle: $h$

Calculation:

The area of rectangle is given as:

  $A=xy$

Put the given data:

  $560=xy$

  $xy=560$  ... (1)

  $y=\frac{560}{x}$

The volume of rectangular cardboard box:

  $\begin{array}{l}\left(x-8\right)\left(y-8\right)4=960\\ \left(x-8\right)\left(y-8\right)=240\\ x\left(y-8\right)-8\left(y-8\right)=240\\ xy-8x-8y+64=240\end{array}$

Put value $xy=560$ :

  $\begin{array}{l}xy-8x-8y+64=240\\ 560-8x-8y+64=240\\ -8x-8y=-384\\ x+y=48\end{array}$

Put value $y=\frac{560}{x}$ :

  $\begin{array}{l}x+y=48\\ x+\frac{560}{x}=48\\ x^2-48x+560=0\\ \left(x-20\right)\left(x-28\right)=0\\ x=20,28\end{array}$

So, get the value of $y=\frac{560}{x}$ as:

  $y=\frac{560}{x}=\frac{560}{20}=28$

Therefore, length and width of cardboard is calculated as $20$ and $28$ .