Finding the Volume and Surface Area of the Space Shuttle External Fuel Tank

Purpose
Use the disk or shell method and surface area formulas for solids of rotation

The Space Shuttle External Fuel Tank

We will break the tank into three pieces and work on each. Since the middle piece is a cylinder, we can use algebra to find its volume and surface area. If you can't remember the formulas, do a search to find them. We'll use calculus for the nose cone and rear cap.

The approximate dimensions of the tank are 154 feet total length, with a 14 foot radius.
The nose cone is about 30 feet long. The rear cap is about 5 feet tall.

Group assignments:

Additional Notes for nose cone groups I and II
Nose cone cross sections come in many shapes - elliptical, parabolic, conical, ...
We will model this as a "Power Series" nose cone.

The formula for the cross section of the nose is: $y=R\left(\frac{x}{L}\right)^n$

where R is the radius of the cone, L is the length of the cone and n=0.5 for our cone.

Graph this on your grapher to see the shape for 0 < x < L. At values of n > about 0.7, the tip becomes sharp (see Wikipedia). You can check this by graphing this function for various values 0 < n < 1.

Write your nose cone y function

and sketch your y graph here (show scales):

With this nose cone function we are ready for the following:

Group 1 - Volume of the nose cone

Using your nose cone function, use calculus to find the volume of the nose cone in cubic feet. Decide on the disk or shell method. Your limits are important.

Write your integral and solution here:

One way to see if you are approximately correct is to have someone in the group calculate the volume of a cone using algebra. You answer should be a little larger.

Group II - Surface Area of the Nose Cone

Using your nose cone function, use calculus to find the surface area of the nose cone in square feet using the surface area integral. Your limits are important.

Write your integral and solution here:

One way to see if you are approximately correct is to have someone in the group calculate the surface area of a cone using algebra. You answer should be a little larger.

Additional Notes for rear cap groups III and IV
The radius of the rear cap is not 14 feet! The rear cap is part of a larger sphere. We will use the dimensions given and this formula to find the radius r of the sphere the cap is contained in:

r=\frac{a^2+h^2}{2h}

Knowing the radius of the sphere, we can write the formula for the circular cross section containing the cap.

Write it here: Solve this circular equation for your rear cap y function:

Sketch your y graph here (show scales):

With this rear cap function we are ready for the following:

Group III - Volume of the rear cap

Using your rear cap function, use calculus to find the volume of the rear cap in cubic feet. Decide on the disk or shell method. Your limits are important.

Write your integral and solution here:

One way to see if you are approximately correct is to have someone in the group calculate the volume of a cone using algebra. You answer should be a little larger.

Group IV - Surface Area of the rear cap

Using your rear cap function, use calculus to find the surface area of the rear cap in square feet using the surface area integral. Your limits are important.

Write your integral and solution here:

One way to see if you are approximately correct is to have someone in the group calculate the surface area of a cone using algebra. You answer should be a little larger.

All Groups - Total Volume and Surface Area

1. Sum the volumes of the 3 pieces to get the total volume of the shuttle external fuel tank.

2a. Sum the surface areas of the 3 pieces to get the total surface area of the shuttle external fuel tank.

b. Why don't they paint the external fuel tank?
Assuming paint weighs about 0.05 lb/ft², how much would the paint on the tank weigh if painted?

(The actual figure is about 600 lb)