# If the base and width of a box remain the same, then the volume is proportional to the height. If...

## Question:

If the base and width of a box remain the same, then the volume is proportional to the height. If one such box is 3 inches tall and another is 24 inches tall, how do their volumes compare?

## Proportionality

In a function, proportionality explains the dependency of one variable to another.

For example: {eq}y=3x^2 {/eq} is a function of {eq}x {/eq}.

It is clear that the value of {eq}y {/eq} depends on the value of {eq}x {/eq}.

Proportionality defines how the value of {eq}y {/eq} is responsive to the value of {eq}x {/eq}.

In the example, {eq}y {/eq} is directly proportional to the square of {eq}x {/eq}, which means that value of {eq}y {/eq} increases with the increase in the value of {eq}x {/eq} and decreases with the decrease in the value of {eq}x. {/eq}

The volume is proportional to the height.

$$V \propto h$$

Taking k as the proportionality constant

$$V =k h$$

At h = 3 inches

$$V_3 = 3k$$

At h = 24 inches

$$V_{24} = 24k$$

$$Ratio\ of\ V_3\ to\ V_{24} = \dfrac{V_3}{V_{24}}$$

$$\dfrac{V_3}{V_{24}}= \dfrac{3k}{24k}$$

$$\dfrac{V_3}{V_{24}}= \dfrac{1}{8}$$

$$V_{24} = 8*V_3$$

The volume {eq}V_{24} {/eq} for the height {eq}24\ inches {/eq} is {eq}8 {/eq} times more than the volume {eq}V_{3} {/eq} for the height {eq}3\ inches. {/eq} 