SOLUTION: Jula is constructing a pot to hold her precious Mathangias. The pot is square based rectangular prism. The surface area on the outside(not inside) is 500 cm2 . Because Jual is goi

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Question 1163357: Jula is constructing a pot to hold her precious Mathangias. The pot is square based rectangular prism. The surface area on the outside(not inside) is 500 cm2
. Because Jual is going to put a plant in the pot, there is no top to the rectangular prism.
a. Determine the maximum volume of the planter pot.
b. What are the dimensions of the candle holder with the maximum volume?
Thank you.
Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
. 
Let x be the size of the square base and h be the height of the pot, in centimeters.


Then the volume of the pot is

     V = x%5E2%2Ah.                           (1)


The surface are of this pot, which has no top, is

      A = x%5E2+%2B+4xh = 500 cm^2.            (2)


So, we want to find optimal values of "x" and "h" to maximize the volume (1) at given restriction (2) on surface area.


From (2), we have 

     h = %28500+-+x%5E2%29%2F%284x%29 = 125%2Fx+-+x%2F4.            (3)


Substitute it into (1) to get

     V = x%5E2%2A%28125%2Fx+-+x%2F4%29%29 = 125x - x%5E3%2F4.     (4)


Now we need to find a maximum value for V in formula (4) considering the volume  as a function of "x" only.


For it, take the derivative of  V(x) and equate it to zero

     V'(x) = 125 - %283x%5E2%29%2F4 = 0.


It implies

     500 = 3x^2

     x^2 = 500%2F3

     x = sqrt%28500%2F3%29 = 12.91 cm (approximately.


Then

     h = (see formula (3)) = 125%2Fx+-+x%2F4 = 125%2F12.91+-+12.91%2F4 = 6.45 cm.



So, the problem is just solved.

Optimal dimensions are: the square base size of 12.91 cm and the height of 6.45 cm.



The maximum volume is  V%5Bmax%5D = 12.91%5E2%2A6.45 = 1075 cm^3  (approximately).

Solved.