The Volume of an Auxiliary Fuel Tank for a Helicopter?

In summary, The conversation discusses the design of an auxiliary fuel tank for a helicopter and the calculations involved. The tank is generated by revolving a given equation around the x-axis and the volume is calculated using an integral. The tank holds approximately 17 cubic feet of fuel, which can extend the helicopter's range by 254.354 miles if it gets 2 miles to the gallon.
  • #1
acdougla17
1
0
I think I did this right but I want to make sure since it is an extra credit problem and I could use the points. If I did something incorrect, I don't want an answer, just point me in the right direction so I can work it out myself.

You are designing an auxiliary fuel tank that will fit under a helicopters fuselage to extend its range. The shape of the tank is generated by revolving y = 1 - (x^2/16), -4<= x => 4 around the x-axis (dimensions in feet).

a) How many cubic feet of fuel will the tank hold (to the nearest cubic foot)?
I set up the integral as 2*pi∫(0 to 4) 1- (x^2/16) dx
My answer was 17 cubic feet

b)A cubic foot holds 7.481 gal. If the helicopter gets 2mi to the gallon, how many additional miles can it fly with the new tank?
My answer 254.354 miles
 
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  • #2
Your answer to (b) is right given your answer to part (a), but you didn't calculate the volume correctly.
 

What is the disk method for finding volume?

The disk method is a technique for finding the volume of a solid of revolution by integrating the areas of disks or cylinders that make up the solid.

What is the formula for calculating volume with the disk method?

The formula for calculating volume using the disk method is V = ∫πR(x)^2dx, where R(x) is the radius of the disk at a given point x.

How do you choose the correct limits of integration when using the disk method?

To choose the correct limits of integration, you need to consider the boundaries of the solid of revolution and the axis of rotation. The limits of integration should encompass the entire solid and be parallel to the axis of rotation.

Can the disk method be used for any shape?

The disk method can be used for any shape that can be rotated around a given axis, such as a circle, square, or even more complex shapes.

How does the disk method differ from the shell method?

The disk method and the shell method are both techniques for finding the volume of a solid of revolution, but they use different shapes (disks vs. cylinders) and have different formulas (πR(x)^2dx vs. 2πxhdx). The shell method is typically used when the axis of rotation is vertical, and the disk method is used when the axis of rotation is horizontal.

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