Chapter 8.4, Problem 27MCQ

To determine

To write the expression for volume of the box.

Answer to Problem 27MCQ

  $4w^3+14w^2+10w$

Explanation of Solution

Given:

Length of the box is $2$ inches more than twice the width and height is $3$ inches more than the length.

Formula used:

Distributive Law:

  $x\left(y\pm z\right)=x\cdot y\pm x\cdot z$

Volume of a rectangular prism $=w\times h\times l$ .

Calculation:

Let the length of the box $l$ , width of the box be $w$ and height of the box be $h$ .

Since length of the box is $2$ inches more than twice the width,

  $\Rightarrow l=2w+\mathrm{2.......}(i)$

Also the height of the box is $3$ inches more than the length,

  $\Rightarrow h=l+\mathrm{3.......}(ii)$

Putting value of $l$ from ( i ) into ( ii )

  $\begin{array}{l}\Rightarrow h=2w+2+3\\ \Rightarrow h=2w+5\end{array}$

Now,

Volume of the box $=w\times h\times l$

  $\begin{array}{l}=w\times \left(2w+5\right)\times \left(2w+2\right)\\ =w\left(2w+5\right)\left(2w+2\right)\\ =w\left(2w\cdot 2w+2w\cdot 2+5\cdot 2w+5\cdot 2\right)\\ =w\left(4w^2+4w+10w+10\right)\\ =w\left(4w^2+14w+10\right)\\ =4w^3+14w^2+10w\end{array}$       [ $\because x\left(y\pm z\right)=x\cdot y\pm x\cdot z$ ]

Conclusion:

Therefore, volume of the box in terms of width is $4w^3+14w^2+10w$ .