<PRE><B>Question 1137932</B>
&lt;br&gt;
This is an excellent problem for you to learn something from, if you do the work yourself.  There is some nice calculus; and there is a lot of ugly arithmetic with expressions like cube root of pi.&lt;br&gt;
I will outline a path to the solution of the problem and hope you will choose to fill in the details.  Or, undoubtedly another tutor will see the problem and give you a full solution; and you can use that if you don&#39;t care to challenge yourself to solve the problem yourself.&lt;br&gt;
The volume of the cylinder is 24:&lt;br&gt;
{{{(pi)(r^2)(h) = 24)}}} (1)&lt;br&gt;
The area of the bottom of the cylinder is {{{(pi)(r^2)}}}; the area of the sides is {{{2(pi)(r)(h)}}}.&lt;br&gt;
With the material for the bottom of the cylinder costing $0.15 per square meter and the material for the sides costing $0.05 per square meter, the cost of producing the cylinder is&lt;br&gt;
{{{.15((pi)(r^2))+.05(2(pi)(r)(h))}}} (2)&lt;br&gt;
Steps for you to take:
1. Solve equation (1) for h in terms of r
2. Substitute into equation (2) to give you a cost function in terms of r only.
3. Find the derivative of the cost function.
4. Set the derivative equal to 0 and solve to find the value of r that minimizes the cost.&lt;br&gt;
So you can check your work, I will tell you that the value of r that minimizes the cost is&lt;br&gt;
{{{2/((pi)^(1/3))}}}&lt;br&gt;
Next step for you:
5. Find an expression for h by substituting this value of r in the equation from step 1. above.&lt;br&gt;
I will tell you that the expression for h is very similar to the expression for r, having a constant in the numerator and a cube root of pi in the denominator.&lt;br&gt;
Last step:
6. Evaluate the cost function for the values of r and h you have found.&lt;br&gt;
With both r and h having the form of a constant divided by the cube root of pi, both terms in the cost function (equation (2) above) will be of the form some constant times the cube root of pi.  When evaluated, that minimum value of the cost function is indeed $2.64 when rounded to the nearest cent.</PRE>