SOLUTION: You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 410 cm^3, what values of h and r will min

Question 969172: You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 410 cm^3, what values of h and r will minimise the total surface area (including the top and bottom faces)? Give your answers correct to 2 decimal places as a list [in brackets] of the form: [ h, r ]
for constants h (height), r (radius), in that order.
x =
Found 2 solutions by Theo, Fombitz:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i couldn't find any other way to do it without using calculus, so i solved it by graphing.

i got r = 4.025946 and h = 8.051894821.

with those values, surface area = 2 * pi * r * h + 2 * p * r^2 = 305.51822

the graph of the equation for surface area looks like this.

the formula for surface area used was:

y = 2 * pi * x * (410/(pi * x^2)) + 2 * pi * x^2

y represents the surface area.
x represents the radius.
h is equal to (410 / (pi * r^2))

this was derived by solving for h in the volume formuls of v = pi * r^2 * h.

when v is equal to 410, this formula became 410 = pi * r^2 * h.

this led to h = (410 / (pi * r^2)).

the formula could have been simplified further but i left it the way it was so you could see the value of h in terms of r stand out more clearly.

the area on the graph of interest is when x and y are both greater than 0.

Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!

From the volume,

Substitute into the surface area equation,

Now surface area is only a function of the radius.
Take the derivative and set it equal to zero to find a min.

Then,

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