Question

z varies directly as square-root of x and inversely as y. If z = 147 when x = 16 and y = 6, find z if x = 25 and y = 4. (Round off your answer to the nearest hundredth.)

Answer 1

From the question

`\begin{gathered} z\propto\sqrt[]{x}\text{ and } \\ z\propto(1)/(y) \\ \text{combining we have } \\ z\propto\frac{\sqrt[]{x}}{y} \\ \propto\text{ is a proportionality constant } \end{gathered}`

Removing the proportionality sign and introducing a constant, we have

`\begin{gathered} z=k*\frac{\sqrt[]{x}}{y} \\ z=\frac{k\sqrt[]{x}}{y} \end{gathered}`

Making k the subject, we have

`\begin{gathered} z=\frac{k\sqrt[]{x}}{y} \\ k\sqrt[]{x}=yz \\ k=\frac{yz}{\sqrt[]{x}} \end{gathered}`

Substituting the initial values of z, x, and y, we have

`\begin{gathered} k=\frac{yz}{\sqrt[]{x}} \\ k=\frac{6*147}{\sqrt[]{16}} \\ k=(882)/(4) \\ k=220.5 \end{gathered}`

The relationship becomes

`z=\frac{220.5\sqrt[]{x}}{y}`

Substituting the second values of x and y into the equation for the relationship, we have

`\begin{gathered} z=\frac{220.5\sqrt[]{x}}{y} \\ z=\frac{220.5\sqrt[]{25}}{4} \\ z=(220.5*5)/(4) \\ z=(1,102.5)/(4) \\ z=275.625 \end{gathered}`

Hence the answer is 275.63 to the nearest hundredth