Question Video: Finding the Average Rate of Change of the Surface Area of an Expanding Sphere Mathematics • Second Year of Secondary School

Video Transcript

A spherical balloon preserves that shape as it expands. Determine the average rate of change of its surface area when its radius changes from 49 centimeters to 119 centimeters.

We’re given a spherical balloon with initial radius 49 centimeters which retains its shape as it expands to a sphere of radius of 119 centimeters. And we’re asked to determine the average rate of change of its surface area as it expands. And in order to find this average rate of change, we need to know two things. The first is the surface area of a sphere, which is four 𝜋𝑟 squared, where 𝑟 is the radius. And the second is the average rate of change of a function 𝑓 of 𝑥 from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏. And that’s equal to 𝑓 at 𝑥 is equal to 𝑏 minus 𝑓 at 𝑥 is equal to 𝑎 over 𝑏 minus 𝑎.

In our case, our function is the surface area, which is actually four 𝜋𝑟 squared, and therefore a function of 𝑟 not 𝑥 since 𝑟 is the variable. So, we have 𝑓 of 𝑟 is equal to four 𝜋𝑟 squared. Our starting value is 𝑟 is equal to 49 centimeters, and this is 𝑎 in our average rate of change function. And our sphere expands to a sphere of radius of 119 centimeters, which is 𝑏 in our average rate of change function. Our average rate of change is, therefore, 𝑓 evaluated at 𝑟 is 119 minus 𝑓 evaluated at 𝑟 is 49 divided by 119 minus 49.

Now remember, 𝑓 is equal to four 𝜋𝑟 squared. We have four 𝜋 times 119 squared minus four 𝜋 times 49 squared over 119 minus 49. We have a common factor of four 𝜋 in our numerator, which we can take outside of our bracket. And our denominator evaluates to 70. And so, we have four 𝜋 over 70 times 119 squared minus 49 squared. Evaluating 119 squared and 49 squared, this gives us four 𝜋 over 70 times 14161 minus 2401. And that’s four 𝜋 over 70 times 11760. And in its simplest form, this is 672𝜋. Our units are centimeters squared per centimeter, that is, the rate of change of the area per unit length.

And so, the rate of change of the surface area of a spherical balloon as the radius changes from 49 to 119 centimeters is 672𝜋 centimeters squared per centimeter.