Our goal is to decide what would affect the volume of a cone more, doubling its radius or doubling its height. Before we decide, let's remember the formula for the volume of a cone.
V = 1/3Ď€ r^2 h
Here r is the radius of the base of the cone, and h is its height. See that the exponents of the variables r and h vary. We can highlight them to see it better.
V = 1/3Ď€ r^2 h^1
In the formula r is raised to the power of 2, and h is raised to the power or 1. This is why they affect the volume differently. In general, the greater the exponent, the more impact the variable has. To see it better, let's write an example.
Example
Consider a cone with a radius of 3 centimeters and a height of 3 centimeters. We can calculate its volume using the formula we mentioned earlier.
The volume of the cone is about 28.26 cm^3. Now, we are going to double the radius and the height to see how this affects the total volume of the figure. Let's start by doubling the radius, which is now 2(3)=6 centimeters.
The volume of the cone is now 113.10 cm^3, which is 113.1028.27 =4 times more than the initial volume of the cone. Now let's double the height and then substitute the new value into the equation.
The second volume is 56.552 cm^3, which is 56.5528.27 =2 times more than the initial volume of the cone. This confirms our suspicions. Doubling the radius has a greater effect.
