# Suppose that S varies directly as the 2/5 power of T, and that S=8 when T=32. Find S when T=243.

## Question:

Suppose that {eq}S {/eq} varies directly as {eq}T^{2/5} {/eq}, and that {eq}S=8 {/eq} when {eq}T=32 {/eq}. Find {eq}S {/eq} when {eq}T=243. {/eq}

## Direct Variation:

Two variables vary directly if an increase/decrease in one variable causes an increase/decrease in the other variable. The ratio of two variables that are directly proportional is equal to a constant, which is referred to as the constant of variation.

If {eq}S {/eq} varies directl as {eq}T^{2/5} {/eq}, we can write this as:

• {eq}S\propto T^{2/5} {/eq}

Removing the proportionality sign and adding a proportionality constant, we have:

• {eq}S = kT^{2/5} {/eq}

If {eq}S = 8 {/eq} when {eq}T = 32 {/eq}, then:

• {eq}8 = k(32)^{2/5} {/eq}
• {eq}8 = k(4) {/eq}
• {eq}k = \dfrac{8}{4} =2 {/eq}

Thus, an equation that relates S to T is:

• {eq}S = 2T^{2/5} {/eq}

Using the above equation, the value of S when T = 243 is equal to:

• {eq}S = 2(243^{2/5}) = \boxed{18} {/eq}