Minimizing optimization

Question

Design a fuel tank that will hold 300 liters of fuel. The shape must be a cylindrical tube with two rounded ends. The side wall (cylinder part) costs $.10 per cm^2 to make while the rounded tops cost $.30 per cm^2 to make. You must find the optimal radius of the tube that will minimize the manufacturing cost.

Answer 1

We first start by converting liters to cubic centimeters. So to do this, we need to multiply 300 by 1000, giving us 300,000.

 

Next, since we need to find out the optimal radius, we need to get the height in terms of the radius. Since the equation for the volume of a cylinder is π(r^2)h=V, We will divide both sides by π(r^2), giving us h=V/(πr^2). Because the volume needs to be 300,000 cubic centimeters, we will replace V with 300,000, giving us h=300,000/(πr^2).

 

Next we will get the equation for the price of the cylinder. We will start with the equation for the surface area, which is 2π(r^2)+2πrh.

 

Next we will multiply the circle parts and the side parts by their respective price per cm^2, which will give us 0.3*2π(r^2)+0.1*2*πr

 

We will then replace h with what we got earlier, giving us 0.3*2π(r^2)+0.1*2πr(300,000/πr^2)

 

Next we will simplify:

0.6*π(r^2)+0.2*300,000πr/(πr^2)

0.6*π(r^2)+60,000πr/(πr^2)

0.6*π(r^2)+60,000/r

 

So our equation is now:     0.6*π(r^2)+60,000/r

 

Now we just need to find the minimum of equation above that is to the right of the y-axis and the x value will be the optimal radius and the y value will be the price of the fuel tank at that radius.

 

The minimum point will be (25.158, 3577.358), meaning that the optimal radius will be 25.158cm.