You are here:

- Home
- Science
- Mathematics
- Advanced Math
- Volume Follow-up

Advertisement

QUESTION: A cylindrical oil tank is laying on it's side on a level rack. A stick is placed through a hole in the upper side, and indicates 1 foot of oil in the tank.

Dimensions of tank: diameter- 10 ft. length- 20 ft.

Calculate number of gallons of oil in the tank.

Thanks.

ANSWER: Look at the tank in cross-section, which is a circle of radius r = 5 ft.

The oil is the segment with height h = 1 ft.

θ = 2·arccos((r - h)/r) = 2·arccos(4/5) ≅ 1.29 radians.

Area of segment = r²(θ-sinθ)/2 ≅ 4.09 ft²

Volume of oil in tank = 20×4.09 = 81.75 ft³

81.75 ft³ × 7.48 gallons/ft³ = 611.5 gallons

[an error occurred while processing this directive]---------- FOLLOW-UP ----------

QUESTION: "θ = 2·arccos((r - h)/r) = 2·arccos(4/5) ≅ 1.29 radians.

Area of segment = r²(θ-sinθ)/2 ≅ 4.09 ft²"

Please explain this part of your solution in more detail. Thanks.

It's all chord math. The basic variables are

s = Arc Length

c = Chord Length

r = Radius

h = Height (outside radius to chord midpoint)

θ = Angle (in radians)

d = Apothem (distance from chord to center of circle)

K = Area of segment (chord to outside)

Using any two of these variables, you can derive the others.

http://www.flickr.com/photos/dwread/12926333565/

You were given r = 5 and h = 1, so d = r - h = 4.

To determine θ, look at the right triangle:

http://www.flickr.com/photos/dwread/12926880404/

You can see that cos(θ/2) = d/r, so θ = 2·arccos(d/r).

area of the sector = (πr²)·θ/(2π) = r²θ/2

Note that sin(θ/2) = c/(2r).

By the double-angle formula, sin(θ/2) = sinθ/(2cos(θ/2)) = r·sinθ/(2d).

area of the right triangle = (½)(c/2)d

= (½)(r²sinθ/(2d))d

= r²sinθ/4

area of the oil segment = (area of sector) - 2(area of right triangle)

= r²θ/2 - 2(r²sinθ/4)

= r²θ/2 - r²·sinθ/2

= r²(θ - sinθ)/2

- Add to this Answer
- Ask a Question

Rating(1-10) | Knowledgeability = 10 | Clarity of Response = 10 | Politeness = 10 |

Comment | Fast and courteous response !!! |

Advanced Math

Answers by Expert:

I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.

I tutor students (fifth through twelfth grades) and am a Top Contributor on Yahoo!Answers with over 24,000 math solutions.**Publications**

Co-author of An Outline of Scientific Writing: For Researchers With English as a Foreign Language**Education/Credentials**

I have a Bachelor's degree in Applied Mathematics from the University of California at Berkeley.**Past/Present Clients**

George White Elementary School.
Homework Help program at the Ridgewood Public Library, Ridgewood, NJ.
Individual students.