What should the radius of the new tank be?

Question

People who live in isolated or rural areas have their own tanks of natural gas to run appliances like stove, washers, and water heaters. These tanks are made in the shape of a cylinder with hemispheres on the ends.The Insane Propane Tank Company makes tanks with this shape, in different sizes. The cylinder part of every tank is exactly 10 feet long, but the radius of the hemispheres,r, will be different depending on the size of the tank. The company wants to double the capacity of their standard tank, which is 6 feet in diameter. What should the radius of the new tank be?

Answer 1

The problem is asking us to consider this shape tank...I will call it a SILO, a cylinder with a hemispherical top...Lets compute the volume of this solid...

 

The cylinder has volume base times height...the height is fixed at 10ft, and the base is given by the formula for area of a circle with radius r, as \pi*r^2...to the cylinder has volume 10\pi(r^2)...The hemispherical top has volume equal to half the volume of a sphere, so that is 4\pi(r^3)/6...putting these together, the SILO has volume V(r)=10\pi(r^2)+4\pi(r^3)/6.

 

Given the diameter is currently 6ft, we have an initial radius of r_0=3, and a SILO volume of 90\pi+18\pi=108\pi...

 

If we wish to double this volume we need to find a new r (larger than 3) for which the volume given by the function V(r) above will be 216\pi...

 

setting V(r)=216\pi=10\pi(r^2)+4\pi(r^3)/6, will allow us to perform algebra on this equation and solve for r.  Alternatively we can graph this function to find the solution.

 

Algebraically:  216\pi=10\pi(r^2)+4\pi(r^3)/6 

so 0=r^3+15r^2-324,

 

Plugging in values for r=4 (-20) and r=5 (176) to the right hand side, tells us immediately there is a solution between r=4 and r=5...so we can graph this and find the solution numerically.  Another approach is to try and factor this cubic polynomial in r and find the solution that way.

 

Hope this helps!

Answer 2

From the information you provided,

 

I believe that the answer would be 4.24 feet. 

 

If the height of every cylinder that the company produces is 10 ft and the volume of a cylinder is found by the equation, V= πr²h, then we can find that the volume of the standard tank is V=π(3²)(10)=282.74.

 

If we want to double this, the new volume will be 282.74*2=565.48 ft³. We need to find the radius that makes this true, so rearranging the formula we have, r=√[(V)/(π*h)]=√[565.48/(π*10)]=4.24 ft