# Suppose z varies inversely with t and that z=6 when t=8. What is the value of z when t=3?

Apr 18, 2018

$\text{ }$
color(red)(z=16

#### Explanation:

The general form of an Inverse Variation is given by

color(blue)(y = k/x, where color(blue)(k is an unknown constant with color(red)(x!=0 and k!= 0

In the equation above, observe that when the value of $\textcolor{b l u e}{x}$ is getting larger and larger, color(blue)(k being a constant, the value of color(blue)(y will be getting smaller and smaller.

This the reason why it is called an Inverse Variation.

For the problem we are solving, the equation is written as

color(brown)(z= k/t, with color(brown)(k being the Constant of Proportionality

It is given that $\textcolor{b r o w n}{z}$ varies inversely as color(brown)(t.

Problem says that color(green)(z=6 when color(green)(t=8

Now you can find $\textcolor{b r o w n}{k}$, the constant of proportionality.

Use

color(green)(z=k/t

$\Rightarrow 6 = \frac{k}{8}$

Rewrite as

$\Rightarrow \frac{6}{1} = \frac{k}{8}$

Cross-multiply to solve for color(green)(k.

$\Rightarrow k \cdot 1 = 6 \cdot 8$

$\Rightarrow k = 48$

color(green)(z=48/t

Next, we need to determine the value of color(green)(z when color(green)(t=3

$z = \frac{48}{3}$, as $t = 3$

rArr color(red)(z= 16