Chapter 6.7, Problem 31E

To determine

To calculate : the height of the rectangular solid.

Answer to Problem 31E

The height is $(3n+2)$.

Explanation of Solution

Given information:

  $\begin{array}{l}\text{Volume of rectangular solid }=12n^3+8n^2-3n-2\\ \text{Length of the solid }=2n+1\\ \text{Width of the solid }=2n-1\end{array}$

Formula used:

  $\text{Volume of the reactangle }=\text{length}\times \text{breadth}\times \text{height}$ .

  $\text{Dividend=Quotient×Divisor+Remainder}$

Calculation:

We have ,

  $\begin{array}{l}{\text{Volume =12n}}^3+8n^2-3n-2\\ \text{Length}=2n+1\\ \text{Breadth}=2n-1\end{array}$

Now,

  $\begin{array}{l}=12n^3+8n^2-3n-2=(2n+1)\times (2n-1)\times \text{height}\\ =\frac{12n^3+8n^2-3n-2}{(2n+1)(2n-1)}=\text{height}\\ \text{=}\frac{12n^3+8n^2-3n-2}{2n(2n-1)+1(2n-1)}=\text{height}\\ \text{=}\frac{12n^3+8n^2-3n-2}{4n^2-2n+2n-1}=\text{height}\\ =\frac{12n^3+8n^2-3n-2}{4n^2-1}=\text{height}\end{array}$

Now, using polynomial division

Step 1.

  $4n^2-1\begin{array}{l}\begin{array}{c}\hfill 3n\\ \hfill \overline{)\begin{array}{l}12n^3+8n^2-3n-2\\ 12n^3+0n^2-3n\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $8n^2-2$

Step 2.

  $4n^2-1\begin{array}{l}\begin{array}{c}\hfill 3n+2\\ \hfill \overline{)\begin{array}{l}8n^2-2\\ 8n^2-2\end{array}}\end{array}\hfill \end{array}$  $\leftarrow \text{Subtract}$

  $0$

  $\begin{array}{l}=12n^3+8n^2-3n-2=\left(3n+2\right)\left(4n^2-1\right)+0\\ =12n^3+8n^2-3n-2=\left(3n+2\right)\left(4n^2-1\right)\end{array}$

Thus, the quotient is $(3n+2)$ and the remainder is $0$.

Therefore, the height of the rectangular solid is $(3n+2)$.