Chapter 11.5, Problem 38PPS

To determine

To calculate: To find the base of the triangle of the prism

Answer to Problem 38PPS

The base of the triangle of the prism is $w+2$

Explanation of Solution

Given information: The volume of a prism with a triangular base is $10w^3+23w^2+5w-2$ and the height of the prism is $2w+1$ and the height of the triangle is $5w-1$

Formula Used:

Volume of prism is given as

  $Volume=Base\times Height$

Calculation:

Given the volume of a prism with a triangular base is $10w^3+23w^2+5w-2$ and the height of the prism is $2w+1$

Area of base of prism is calculated as follows:

  $\begin{array}{l}Areaofbase=\frac{Volume}{Height}\\ Areaofbase=\frac{10w^3+23w^2+5w-2}{2w+1}\end{array}$

Dividing the polynomial with the divisor to derive the area of the base:

  $2w+1\begin{array}{c}\hfill 5w^2+9w-2\\ \hfill \overline{)\begin{array}{l}10w^3+23w^2+5w-2\\ 10w^3+5w^2\\ --\\ ---------\\ 0+18w^2+5w\\ +18w^2+9w\\ --\\ ---------\\ -4w-2\\ -4w-2\\ +-\\ ---------\\ \text{ 0}\end{array}}\end{array}$

Thus, area of base of triangle is $5w^2+9w-2$

Now, length of base of triangle is calculated as follow:

  $\begin{array}{l}Base=\frac{Area}{Height}\\ Base=\frac{5w^2+9w-2}{5w-1}\end{array}$

Dividing the polynomial with the divisor to derive the base:

  $5w-1\begin{array}{c}\hfill w+2\\ \hfill \overline{)\begin{array}{l}5w^2+9w-2\\ 5w^2-w\\ -\text{ + }\\ ---------\\ 0+10w-2\\ +10w-2\\ -\text{ +}\\ ---------\\ \text{ 0}\end{array}}\end{array}$

Thus, base of triangle is $w+2$

Thus, the quotient is $p^2-3p-3+\frac{6}{p-1}$

Conclusion:

Hence, the base of the triangle of the prism is $w+2$