Question

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

Answer 1

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`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 `color(blue)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 `color(brown)z` varies inversely as `color(brown)(t`.

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

Now you can find `color(brown)k`, the constant of proportionality.

Use

`color(green)(z=k/t`

`rArr 6=k/8`

Rewrite as

`rArr 6/1=k/8`

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

`rArr k*1 =6*8`

`rArr k = 48`

Your inverse equation now becomes

`color(green)(z=48/t`

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

`z= 48/3`, as `t= 3`

`rArr color(red)(z= 16`

which is the required answer.

Hope it helps.