100 POINTS A storage tank will have a circular base of radius r and a height of r. The tank can be either cylindrical or hemispherical​ (half a​ sphere). Complete parts​ (a) through​ (e) below.

a. Write and simplify an expression for the ratio of the volume of the hemispherical tank to its surface area​ (including the​ base). For a​ sphere, V = 4/3πr^3 and SA = 4πr^2
What is the volume of the hemispherical tank​ (including the​ base)?
V=
What is the surface area of the hemispherical tank​ (including the​ base)?
SA=

What is the ratio of the volume of the hemispherical tank to its surface​ area, including the​ base? (write an equation)


b. Write and simplify an expression for the ratio of the volume of the cylindrical tank to its surface area​ (including the​ bases).

What is the volume of the cylindrical​ tank?
V=
What is the surface area of the cylindrical tank​ (including the​ bases)?
SA=
What is the ratio of the volume of the cylindrical tank to its surface​ area, including the​ base?


c. Compare the ratios of volume to surface area for the two tanks. Choose the correct answer below.
- The ratio for the cylindrical tank is always smaller.
- The ratios of both tanks are the same.
- The ratio for the cylindrical tank is always larger.

d. Compare the volumes of the two tanks. Choose the correct answer below.
- For any value of​ r, the cylindrical tank will have a smaller volume.
- For any value of​ r, the cylindrical tank will have a larger volume.
- For any value of​ r, both tanks have the same volume.

e. Describe how you used these ratios to compare the volumes of the two tanks.
pick one in the parenthesis

Comparing the (constant or variable part) of each ratio shows that a cylindrical tank will always have (the same, a smaller, or a larger volume) as the hemispherical tank with the same radius.