Chapter 9.5, Problem 30PPS

To determine

To find the value of $b_2$

Answer to Problem 30PPS

the value of $b_2$ is $-105$

Explanation of Solution

Given information:

a varies directly as b and inversely as c

the value of $b_1=15\text{ when }{a}_1=4\text{ and }{c}_1=2$

Calculation:

The values of $b_2$ can be obtained as

First set up a correct proportion for the information given

  $\begin{array}{l}{a}_1=\frac{k{b}_1}{c_1}{\text{ and a}}_2=\frac{k{b}_2}{c_2}\text{ [since, }b\text{ varies directly as }a\text{ so }b\text{ goes in the numerator and}\\ c\text{ varies inversely as }a\text{,so }c\text{ goes in the denominator]}\\ k=\frac{a_1{c}_1}{b_1}\text{ and }k=\frac{a_2{c}_2}{b_2}\text{ [solve for }k\text{]}\\ \frac{a_1{b}_1}{c_1}=\frac{a_2{b}_2}{c_2}\text{ [set the two proportion equal to each other]}\end{array}$

Put the values of $a_1=4$ , $b_1=15$ , $c_1=2$ , $a_2=7\text{ and }{c}_2=-8$ in above equation

  $\begin{array}{l}\frac{\left(4\right)\left(2\right)}{15}=\frac{\left(7\right)\left(-8\right)}{b_2}\\ \left(4\right)\left(2\right)\left(b_2\right)=\left(15\right)\left(7\right)\left(-8\right)\left[\text{cross multiply}\right]\\ 8b_2=-840\left[\text{simplify}\right]\\ b_2=-105\left[\text{divide each side by 8}\right]\end{array}$

Hence,

the value of $b_2$ is $-105$