Question

The volume (V) of a sphere varies directly with the cube of its radius (r). If k is the constant of proportionally, which is the formula for this relationship?

Answer 1

This should be relatively easy, since most of us have learned and used the formula for the volume of a sphere:  V = (4/3)* pi* r^3.

Just combine (4/3)pi into a single value, k, which results in   V = kr^3.

Answer 2

If k is the constant of proportionally, the formula for this relationship is V=kr³.

What is a directly proportional and inversely proportional relationship?

Let there are two variables p and q

Then, p and q are said to be directly proportional to each other if

 p = kq

where k is some constant number called the constant of proportionality.

This directly proportional relationship between p and q is written as

 p∝q  where that middle sign is the sign of proportionality.

In a directly proportional relationship, increasing one variable will increase another.

Now let m and n be two variables.

Then m and n are said to be inversely proportional to each other if

$m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}$

(both are equal)

where c is a constant number called constant of proportionality.

This inversely proportional relationship is denoted by

$m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}$

As visible, increasing one variable will decrease the other variable if both are inversely proportional.

The volume (V) of a sphere varies directly with the cube of its radius (r). Therefore, we can write,

V ∝ r³

Now, To remove the proportionality we need to add a constant of proportionality,

V = kr³

Hence, If k is the constant proportionally, the formula for this relationship is V=kr³.

Learn more about Directly and Inversely proportional relationships:

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