HELP WITH CALC PROJECT!! Pleaaase! Optimization

[equinox]

Hi. I have a calc project that requires me to come up with a hexagonally based can that hold 355 ml. I need to find the lowest cost. Here is what I know...

Cost of sides= 0.01 cents / cm^2
Cost of top= 0.02 cents / cm^2
Cost of bottom= 0.03 cents / cm^2

this is the actual prompt in case. Your team has been given the assignment of submitting a packaging plan for a new product. Prism Pop is to be sold in hexagonally based cans, each holding 355 milliliters of soda. The management prefers plans that lower the cost. The material for the sides costs 0.01 cents per square centimeters. The material for the bottom costs 0.03 cents per square centimeter. The material for the top costs 0.02 cents per square centimeter.

Im so lost I don't even know where to begin so detailed steps would be greatly appreciated!

 

[stapel]

Your team has been given the assignment of submitting a packaging plan for a new product. Prism Pop is to be sold in hexagonally based cans, each holding 355 milliliters of soda. The management prefers plans that lower the cost. The material for the sides costs 0.01 cents per square centimeters. The material for the bottom costs 0.03 cents per square centimeter. The material for the top costs 0.02 cents per square centimeter.

Im so lost I don't even know where to begin so detailed steps would be greatly appreciated!

Try starting with stuff you learned back in algebra and geometry: What is the formula for the area of a regular hexagon? What is the formula for the volume of a regular hexagon? For the surface area? How do cubic centimeters relate to milliliters? How does the stated volume relate to the formulas?

And so forth.

 

[HallsofIvy]

The first thing you will need is a formula for the volume of a "hexagonally based can". That will be, of course, the height times the base. The base is a hexagon- you don't say "regular hexagon" but I assume that is what you mean. You should know that, by drawing lines from the center to each vertex, a hexagon can be thought of as 6 identical equilateral triangles. If we take the base to have 6 sides of length "s" then we have 6 equilateral triangles with side length s so the area of the base is 6 times the area of each triangle. Each triangle, itself, can be thought of as two right triangles, with one side as hypotenuse and half the base a leg. If the hypotenuse is s and one leg has length s/2 then, by the Pythagorean theorem. the other leg has length $\displaystyle \sqrt{s^2- s^/4}= s\frac{\sqrt{3}}{2}$. That is the "altitude" of each equilateral triangle so the area of each is "1/2 base times length"= $\displaystyle s^2\frac{\sqrt{3}}{4}$. The area of the base is 6 times that: $\displaystyle s^2\frac{3\sqrt{3}}{2}$ and the volume of the hexagonal based can is that area times its height, $\displaystyle s^2h\frac{3\sqrt{3}}{2}= 355$.

The cost of the base is $\displaystyle 0.03\left(s^2\frac{3\sqrt{3}}{2}\right)$, the cost of the top is $\displaystyle 0.02\left(s^2\frac{3\sqrt{3}}{2}\right)$. Each of the 6 sides is a rectangle with width s and height h so area sh and cost $\displaystyle 0.01 sh$. Find the total cost of the can in terms of s and h and use the fact that $\displaystyle s^2h\frac{3\sqrt{3}}{2}= 355$ to reduce to a single variable, either s or h.

 

[equinox]

Thanks for the help! I'm still a little stuck on the last part though. I've tried solving for the total cost but not sure. do i just solve for the cost of bottom, top, and sides separately and add them all together? Also, am i essentially solving for "s" here? For example, the cost of the base is [FONT=MathJax_Main]0.03[/FONT][FONT=MathJax_Size2]([/FONT][FONT=MathJax_Math]s[/FONT][FONT=MathJax_Main]^2*3sqrt3/2[/FONT][FONT=MathJax_Size2]) , how would I go about solving for "s"? If i use a cost function, it would be [/FONT][FONT=MathJax_Main]0.03[/FONT][FONT=MathJax_Size2]([/FONT][FONT=MathJax_Math]s[/FONT][FONT=MathJax_Main]^2*3sqrt3/2[/FONT][FONT=MathJax_Size2]) +[/FONT][FONT=MathJax_Main]0.02[/FONT][FONT=MathJax_Size2]([/FONT]s^2*3sqrt3/2) + (.01sh). I have tried setting this equation equal to zero but dont know how to solve for the cost.

 

[stapel]

I've tried solving for the total cost but not sure. do i just solve for the cost of bottom, top, and sides separately and add them all together?

You're needing to find the minimal cost of the can, not the minimal costs of each separate part (which would likely be "don't use any of this at all" for each portion).

Please reply showing what you have tried. Thank you!