Calculus question help

Question

The Betty Moore Company requires that its corned beef hash containers have a capacity of 40 in^3, be right circular cylinders, and be made of a tin alloy. Find the radius and height of the least expensive container that can be made if the metal for the side and bottom costs 4¢/in.2 and the metal for the pull-off lid costs 1 ¢/in^2. (Hint: Let the radius and height of the container be r and h in. respectively. Then, the volume of the container is πr2h=40, and the cost is given by C(r,h) = 8πrh + 5πr2.) (Give your answers correct to 2 decimal places.)

please help!!
find r=
and h=
thanks!

Answer 1

The formula for the volume.

 

V = πr²h

 

Therefore,

 

πr²h = 40                 eq1

 

 

The formula for surface area.

 

SA = 2πr² + 2πrh

 

If we consider the cost in the surface area, we obtain the cost formula

 

C = 0.04(πr² + 2πrh) + 0.01πr²

C = 0.04πr² + 0.08πrh + 0.01πr²

C = 0.05πr² + 0.08πrh                            eq2

 

Your formula was correct, but the number should be decimal values because you want your cost to be in the form of dollars and cents.  For example, 5 cents is the same as $0.05

 

Substitute the eq1 into equ2 so that eq2 is in terms of only r.

 

C = 0.05πr² + 0.08πr(40/πr²)

C = 0.05πr² + 3.2r^-1

 

Now all you have to do is set the derivative of C equal to zero.  Then solve for r.  When you have solved for r, evaluate the derivative around the point.  You want the derivative to change from negative to positive.  Also, your r should be positive.  That will be the value of r for least cost.

 

Once you have your value of r, plug it into the volume equation to solve for h.