Chapter 9.5, Problem 24PPS

To determine

To find the value of $z_2$

Answer to Problem 24PPS

the value of $z_2$ is $-6$

Explanation of Solution

Given information:

x varies directly as y and inversely as z

the value of $z_1=20\text{ when }{x}_1=6\text{ and }{y}_1=14$

Calculation:

The values of $z_2$ can be obtained as

First set up a correct proportion for the information given

  $\begin{array}{l}{x}_1=\frac{k{y}_1}{z_1}\text{ and }{x}_2=\frac{k{y}_2}{z_2}\text{ [since, }x\text{ varies directly as }y\text{ so }y\text{ goes in the numerator and}\\ x\text{ varies inversely as }z\text{,so }z\text{ goes in the denominator]}\\ k=\frac{x_1{z}_1}{y_1}\text{ and }k=\frac{x_2{z}_2}{y_2}\text{ [solve for }k\text{]}\\ \frac{x_1{z}_1}{y_1}=\frac{x_2{z}_2}{y_2}\text{ [set the two proportion equal to each other]}\end{array}$

Put the values of $x_1=6$ , $y_1=14$ , $z_1=20$ , $x_2=10{\text{ and y}}_2=-7$ in above equation

  $\begin{array}{l}\frac{\left(6\right)\left(20\right)}{14}=\frac{\left(10\right)\left(z_2\right)}{-7}\\ \left(6\right)\left(20\right)\left(-7\right)=\left(14\right)\left(10\right)\left(z_2\right)\left[\text{cross multiply}\right]\\ -{840}_{}=140z_2\left[\text{simplify}\right]\\ z_2=-6\left[\text{divide each side by 140}\right]\end{array}$

Hence,

the value of $z_2$ is $-6$