Packaging: The Optimal Form HELP!

  • #1

Homework Statement



1. Create a table of values for the dimensions of a cylinder with a volume of 49.54 cubic inches. Does it appear that the cleanser container minimizes surface area?
2. Suppose you are designing a coffee creamer container that has a volume of 48.42 cubic inches. Use the equations for the surface area of a cylinder and the volume of a cylinder to develop an equation relating the radius r and surface area S. S=2pi(r)^2+2pi(r)(h). V=pir^2h
3. Repeat Question 2 for each of the other containers in the table. Use a graphing utility to plot each equation. Determine whether the radius of each container is larger than, smaller than, or equal to the optimal radius.
4. Suppose in order to fit more writing on the cylinder you want to maximize the surface area of a cylinder that holds 49.5 cubic inches. Can you do this? Explain.

Product-Radius-Height-Volume
Coffee Creamer 1.5 6.85 48.42
Cleanser 1.45 7.5 49.54
Coffee 1.95 5.2 62.12
Pineapple 2.1 6.7 92.82
Frosting 1.63 3.6 30.05
Soup 1.3 3.8 20.18
Tomato 1.95 4.4 52.56
Baking Powder 1.25 3.65 17.92

Homework Equations


SA=SA=2πr 2 + 2πrh


The Attempt at a Solution


1. (1) For the heights of the container the equation is: h=49.54/r2
49.54 is the volume of the cylinder and r is the radius of the cylinder. To find height, the equation for the volume of the cylinder (V=πhr2) is made to solve for h.
(2) To find the surface area, the equation, SA=2πr 2 + 2πrh is needed.
(3) Since h=V V/πr 2, plug it into the equation SA=2πr 2 + 2πrh so it becomes SA=2πr 2 + 2πr(V/πr 2) àSA=2πr 2 + 2V/r
(4) Find the derivative: SA’=2π(2r+(-49.54)/r2)
(5) Solve for r. r=3√24.77=2.915in
No, the cleaner container does not minimize the surface area as the radius of 2.915 inches is larger than the original 1.45 inches.
2.. An equation that can relate the radius r to the surface area S is SA=2π(r2+(48.42/r) since
(1) h=48.42/r2 comes from V=πhr2 when you solve for h.
(2) The equation for surface area is SA=2π(r2+hr)
(3) Plug the h into the equation.

3. Coffee Creamer SA=2π(r2+(48.42/r)= 78.6969 in.2
Cleanser SA=2π(r2+(49.54/r)= 81.5400 in.2
Coffee SA=2π(r2+(62.12/r)= 87.6033 in.2
Pineapple juice SA=2π(r2+(92.82/r)= 116.1133 in.2
Frosting SA=2π(r2+(30.05/r)= 53.5635 in.2
Soup SA=2π(r2+(20.18/r)= 41.6575 in.2
Tomato puree SA=2π(r2+(52.56/r)= 77.8015 in.2
Baking Powder SA=2π(r2+(17.92/r)= 38.4845 in.2

4. No, you cannot because without knowing the limit for the radius and height before you can maximize the surface area. If not, the surface area will keep on maximizing. This would lead to, in real life, the product to be too large for consumer use or not appealing to consumers.

please help me check and help me with the parts I didn't do. Thank you and sorry! My teacher does not teach and I am having trouble understanding concepts.
 

Answers and Replies

  • #2
haruspex
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(3) Since h=V V/πr 2, plug it into the equation SA=2πr 2 + 2πrh so it becomes SA=2πr 2 + 2πr(V/πr 2) àSA=2πr 2 + 2V/r
(4) Find the derivative: SA’=2π(2r+(-49.54)/r2)
Something looks wrong there. There was no factor pi in the right-hand term before differentiation, now there is.
2.. An equation that can relate the radius r to the surface area S is SA=2π(r2+(48.42/r) since
A right parenthesis is missing. Looks like you may have the same error as above.
 

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