Calc optimization: "a rectangular closed box of height h with a length 1 with..."

[kamurphy]

Calc optimization: "a rectangular closed box of height h with a length 1 with..."

1. a rectangular closed box of height h with a length l with 3times its width w has a volume V = 50m3.the bottom and the top side are made of material cost $10/m2 and the remaining sides cost $15/m2.Find l, w and h for which it minimize the cost of the box?
   

 

[Deleted member 4993]

2. a rectangular closed box of height h with a length l with 3times its width w has a volume V = 50m3.the bottom and the top side are made of material cost $10/m2 and the remaining sides cost $15/m2.Find l, w and h for which it minimize the cost of the box?

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[stapel]

A rectangular closed box of height h with a length l with 3 times its width w has a volume V = 50m³. The bottom and the top side are made of material cost $10/m² and the remaining sides cost $15/m². Find l, w, and h for which it minimize the cost of the box?

Please reply with the exact text of the exercise. For instance, I'm fairly certain you wouldn't have been asked to "Find 1", and "with a length 1 with 3 times its width" doesn't make sense in English.

When you reply, please include a clear listing of your thoughts and efforts so far, so we can see where you're getting stuck. Thank you!

 

[HallsofIvy]

staple, it looks to me like that is the letter "l" (ell). not the number 1.

kamurphy, The volume of a box is given by "height times length times width". Since here the length is twice the width, that is "height times 2 times width squared": \(\displaystyle V= 2hw^2\) where h is the height and w is the width. Further, area of each rectangle is "length times width". For the top and side that is \(\displaystyle 2(2w^2)= 4w^2\) square meters. Since the material cost $10 per square meter, how much does that cost? Two of the sides are wh and the other two sides are 2wh each for a total of wh+ 2wh= 3wh. The material for the sides cost $15 per square meter. How much will the sides cost? Your problem is to minimize the total cost subject to the constraint \(\displaystyle V= 2hw^2= 50\). There are several methods for doing that. Which have you learned?