Ok, well the problem states:(adsbygoogle = window.adsbygoogle || []).push({});

A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?

The first step that I took was to draw a picture. I just drew a semicircle with a right circular cylinder right in the middle with the base on the bottom of the hemisphere and the top touching at two points to the rounding of the semicircle. I've attached a crude representation of my drawing.

Then, I defined the volume of this storage tank, V = (1/2)(4/3*pi*r(hemisphere)^3) - (pi*r(cylinder)^2*h(cylinder)). I kept in mind the fact that V is a constant that was given in the problem and that I would use it in my final equation that I would optimize.

I am not sure of this is correct but it seems to me that the metal storage tank would be a hemisphere with a hole that is a right circular cylinder...

Next, I tried to eliminate one of the three variables. I did this by drawing a triangle from the top-right corner of the cylinder to the center of the cylinder and connected it to the right side of the cylinder. I have attached a representation of this too.

I used the pythagorean theorem from there and said that (h/2)^2 = (r(hemi))^2 - (r(cyli))^2. Then I continued to square root both sides to solve for h/2.

I thought that I would need to solve for the surface area of the tank to see how much material would be needed and I write an equation for that as A = (1/2)(4*pi*r(hemi)^2) - (2*pi*r(cyli)^2 + 2*pi*r(cyli)*h). After that I tried to get r(cyli) onto one side so that I could solve for it but I ended up with a rather large and extremely convoluted mess from which I was not able to do so.

Any help would be appreciated. I'm stuck as to the actual representation of the problem...*sigh*...

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# Homework Help: Optimization Metal Tank Problem

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