Chapter 12.1, Problem 40WE

To determine

To find: that by what percent must the height be increases for the volume to remain unchanged.

Answer to Problem 40WE

The height must be increased by $56.25\%$ for the volume to remain unchanged.

Explanation of Solution

Given information:

The length and width of a rectangular solid area each decreases by $20\%$ .

Calculation: to find the increased height percent, equal the volumes before decreasing the length-width and after decreasing the length width,

Let the height increased $x\%$ ,

  $\begin{array}{l}{v}_a=v_b\\ b_a{h}_a=b_b{h}_b\\ \left(l-\frac{20l}{100}\right)\left(w-\frac{20w}{100}\right)\left(h+\frac{xh}{100}\right)=lwh\text{ [substitute]}\\ \left(\frac{80l}{100}\right)\left(\frac{80w}{100}\right)\left(\frac{\left(100+x\right)h}{100}\right)=lwh\text{ [simplify]}\\ \left(\frac{80}{100}\right)\left(\frac{80}{100}\right)\left(\frac{100+x}{100}\right)=1\\ \frac{100+x}{100}=\frac{10000}{6400}\text{ [cross multiplication]}\\ \frac{100+x}{100}=1.5625\text{ [divide]}\\ 100+x=156.25\text{ [multiply]}\\ x=56.25\%\text{ [subtract]}\end{array}$

Thus, The height must be increased by $56.25\%$ for the volume to remain unchanged.