Chapter 8.5, Problem 49HP

To determine

To find the relation of a and g .

Answer to Problem 49HP

a is in combined variation with g

Explanation of Solution

Given information:

The given statement is a varies inversely as b , c varies jointly as b and f , and f varies directly as g .

Calculation :

First write the variations in mathematical form

  $\begin{array}{l}a\propto \frac{1}{b}\\ a=\frac{k_1}{b}\mathrm{...}\left(\text{i}\right)\\ c\propto bf\\ c=k_{2}bf\mathrm{...}\left(\text{ii}\right)\\ f\propto g\\ f=k_{3}g\mathrm{...}\left(\text{iii}\right)\\ \text{from equation }\left(\text{ii}\right)\text{ and }\left(\text{iii}\right)\\ c=k_{2}b{k}_{3}g\left[f=k_{3}g\mathrm{...}\left(\text{iii}\right)\right]\\ b=\frac{c}{k_2{k}_{3}g}\left[\text{simplify}\right]\\ b=\frac{c}{k_2{k}_{3}g}\left(\text{iv}\right)\\ \text{from equation }\left(\text{i}\right)\text{ and }\left(\text{iv}\right)\\ a=\frac{k_1}{\frac{c}{k_2{k}_{3}g}}\\ a=\frac{k_1{k}_2{k}_{3}g}{c}\end{array}$

Hence,

a is in combined variation with g