A soap bubble maintains its spherical shape as it expands. Determine the average rate of change in its surface area when its radius changes from 10 centimeters to 12 centimeters.
We have a sphere of radius 10 centimeters, and we’re asked to find the average rate of change in its surface area when it expands to a sphere of radius 12 centimeters. In order to find the rate of change in its surface area, we need to know two things. The surface area of a sphere 𝐴, which is four 𝜋𝑟 squared, and the average rate of change of a function 𝑓 of 𝑥 on an interval 𝑎, 𝑏. And that’s equal to 𝑓 of 𝑏 minus 𝑓 of 𝑎 all over 𝑏 minus 𝑎. You may also see this written as 𝑓 of 𝑥 plus ℎ minus 𝑓 of 𝑥 over ℎ.
In our case, 𝑓 of 𝑥 is actually our surface area function, 𝐴 of 𝑟. Our interval has an upper bound of 12 and a lower bound of 10 so that 𝑎 is 10 and 𝑏 is 12. And our average rate of change is 𝑎 of 12 minus 𝑎 of 10 over 12 minus 10, which is 𝑎 of 12 minus 𝑎 of 10 over two. To work out the value of this rate of change, we need to find 𝑎 of 12 and 𝑎 of 10. That is the surface area when 𝑟 is 12 and when 𝑟 is 10.
And the surface area when 𝑟 is 12 is four times 𝜋 times 12 squared, which is equal to four 𝜋 times 144. When the radius is 10, the surface area is four times 𝜋 times 10 squared, which is four 𝜋 times 100. If we substitute these now into our average rate of change, we have four 𝜋 times 144 minus four 𝜋 times 100 over two. Where four 𝜋 times 144 is the surface area when the radius is 12 and four 𝜋 times 100 is the surface area when the radius is 10.
Since we have a common factor of four 𝜋, let’s take that outside, which gives us four 𝜋 times 144 minus 100 over two. We can cancel the two. And inside the bracket, we have 44. So, we have two 𝜋 times 44, which is 88𝜋. We mustn’t forget the units, which are centimeter squared per centimeter. The centimeter squared refers to the surface area, and that’s per centimeter, which is the unit of the radius.
So, the average rate of change in the surface area of a bubble when its radius changes from 10 centimeters to 12 centimeters is 88𝜋 centimeter squared per centimeter.