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

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.