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Volume maximization-Calculus

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A shelter for the beach...
A shelter for the beach has a back, 2 square sides and a top. Suppose that 250 square feet of canvas are to be used. Find the smallest dimension of the shelter for which the space inside the shelter (volume) will be maximized

I need to know all the steps to this. Thanks!
Submitted: 13 years ago.Category: Homework
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Answered in 8 minutes by:
4/6/2005
Tutor: Hatice Ozer, Graduate Student replied 13 years ago
Hatice Ozer
Hatice Ozer, Graduate Student
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 we know that


total area is let say one side X


and height is h


total area= 2x^2+4xh=250


then total volume is


x^2*h


so that we know


 h=250-2x^2


   ----------------


        4x


then we will find the volume which is


V=x^2*(250-2x^2)


   -----------------------


              4x


V=(250x-2x^3)/4


we will take the derivative with respect to x and equat to 0


250=6x


x=125/3


so that the base side will be 125/3


 


 


 


 


 

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Customer reply replied 13 years ago
Reply to Hatice Ozer's Post: So what is the smallest dimension? I know it's between 6 and 8 feet but just can't figure out what it is
Tutor: jacket, BS degree in math replied 13 years ago
jacket
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Tutor: RoberttheRocket, Teacher/tutor replied 13 years ago
RoberttheRocket
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 whoa!


don't forget the h = 250 - 2x^2/4.


Since the h = 250 - 2x^2/4x 


then t will equal (250 - 2(6.45497)^2 )/ 4(6.45497) 


=  (250 - 83.333275) / 25.81988 = 6.45497 also


so the shape of the shelter will be a  cube. 


 


Note:  I have a problem with this because the 250 square feet will cover the shelter (the sides, back, and top), but in the calculations, I see all six sides have been used for the surface area. (250 = x^2 + x^2 + 4xh.)


                 

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Tutor: xarqi, Generalist replied 13 years ago
xarqi
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My assessment of the answers above:

Jacket's response is excellent.

RoberttheRocket's point is well made, but does not refer to
Jacket's answer.

Site rules prevent me from commenting openly on the quality of
Hatice Ozer's answer.
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