As part of a real world application of the surface area and volume formulas for spheres, I came up with the following lesson. It is practical and involves the use of the formulas as a step in a process rather than as an end. Students will be helping their uncle, the founder/ owner/ manager of a new gym, World's Gym. As part of the designing process they will be needing to find out two pieces of information.
1. As part of the store front, they will be placing a huge world (sphere) by the entrance to garner attention from those passing by. The owner needs to know how much paint to buy, thereby requiring students to calculate the surface area of the sphere.
2. The owner is also responsible for purchasing some equipment. Of course, there will be a basketball gym in World's Gym so there needs to be somewhere to store the balls. What size cage should the owner purchase?
So, this task requires students to apply the formulas for surface area and volume of a sphere in a more realistic setting. They will also be asked to analyze the effect of doubling the radius on the volume of the sphere and make a formalized conjecture.
Click here for the link to a handout you could give. Feel free to adapt as necessary.