If z varies inversely as w, and z = 40 when w = .06, find z when w = 20.

Question:

If z varies inversely as w, and z = 40 when w = .06, find z when w = 20.

Inverse Variation:

Inverse variation gives a relationship between two variables that vary inversely. That is, a change in one variable leads to an opposite change in the other variable. For variables that are inversely related, their product is equal to a constant, known as the constant of variation.

Given that {eq}z {/eq} varies inversely as {eq}w {/eq}, we can write this as:

• {eq}z\propto \dfrac{1}{w} {/eq}.

Removing the proportionality sign, we have:

• {eq}z = \dfrac{k}{w} {/eq}.

If {eq}z = 40 {/eq} when {eq}w = 0.06 {/eq}, the proportionality constant is equal to:

• {eq}40 = \dfrac{k}{0.06} {/eq}.
• {eq}k = 40\times 0.06 = 2.4 {/eq}.

Thus, an equation that gives the value of z for any given value of w is equal to:

• {eq}z = \dfrac{2.4}{w} {/eq}.

Using the above relation, the value of z when w = 20 is equal to:

• {eq}z = \dfrac{2.4}{20} = \boxed{0.12} {/eq}.