You are here:

Calculus/Propane tank optimization


A company is designing propane tanks that are cylindrical with hemispherical ends. Assume that the company wants tanks that will hold 1000 cubic feet of gas, and that the ends are more expensive to make, costing $5 per square foot, while the cylindrical barrel between the ends costs $2 per square foot. Determine the minimum cost to construct such a tank

Assume the tank has radius r and height x.
The volume is then 2*pi*r^2 on each end and 2*pi*r*x on the sides.
The cost is then 5*2*2*pi*r^2 for both ends and 2*2*pi*r*x for the sides.
That is, 20*pi*r^2 + 8*pi*r*x.

The volume of the tank is (4/3)*pi*r^3 + pi(r^2)x = pi(r^2)(4r/3 + x).
Since the volume is 1000, set 1000 =  pi(r^2)(4r/3 + x) and solve for x.
(1000 - pi(r^2)(4r/3) = pi(r^2)x, so x = (1000 - pi(r^2)(4r/3)/(pi(r^2)).
That works out to be x = 1000/(pi(r^2)) - 4r/3.

Put that back into 20*pi*r^2 + 8*pi*r*x for x.
Simplify out into two terms and reduce r where necessary.

This gives the only variable as r, so differentiate with respect to r.

Set that formula to 0 and solve for r.

Using this value of r, the value of x can be found.


All Answers

Answers by Expert:

Ask Experts




Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).


Experience in the area: I have tutored students in all areas of mathematics since 1980. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC. Well over 2,000 of them have been in calculus.

I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few courses at college a year early.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.

©2016 All rights reserved.