Calculus: Analytic Geometry - optimization problem [Help]

[WilliamTheAnalyst]

I've been trying for the longest to solve these two problems but I just keep getting them wrong and I've no clue where I'm going wrong. This is for my Calculus: Analytic Geometry class, help would be very much appreciated!

1.) A cylinder shaped can needs to be constructed to hold 550 cubic centimeters of soup. The material for the sides of the can costs 0.03 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.07 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

Helpful information:
h : height of can, r : radius of can

Volume of a cylinder: V=πr^2h

Area of the sides: A=2πrh

Area of the top/bottom: A=πr^2

To minimize the cost of the can:
Radius of the can: ___
Height of the can: ___
Minimum cost: ___ cents

2.) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=3-x^2. What are the dimensions of such a rectangle with the greatest possible area?

 

[HallsofIvy]

?? You say "I've been trying for the longest to solve these two problems but I just keep getting them wrong", but you do not show any of your attempts!

Yes, the top and bottom of the can each have area \(\displaystyle \pi r^2\) so together they have area \(\displaystyle 2\pi r^2\) sq cm.. The material for top and bottom cost 0.07 cents per square cm. so what is the cost of top and bottom?

Yes, the area of the sides is \(\displaystyle 2\pi rh\) square cm. The material for the sides cost 0.05 cents per square cm. so what is the cost of the sides?

What is the cost of the entire can, as a function of r and h?

This is "Analytic Geometry", not "Calculus", so you haven't yet learned to deal with "functions of several variables". But you are also told that the volume is 550 cubic cm and, yes, volume is \(\displaystyle \pi r^2h\) so you have \(\displaystyle \pi r^2h= 550\). You can solve that for h as a function of r and replace h in the cost equation by that.

Once you have done that, you will have the cost as a function of r only. What methods have you learned for finding the minimum value of a function like that?