Answered: What should the dimensions of the box…

Name: I am designing a box for my cat. The box will be a rectangular solid w
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Description: I am designing a box for my cat. The box will be a rectangular solid with no top and a length that is twice its width. The box will have a volume of 36ft^3. What should the dimensions of the box be in order to minimize the surface area?

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter5: Factoring Polynomials

Section5.13: Using Factoring To Solve Problems

Problem 27P

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I am designing a box for my cat. The box will be a rectangular solid with no top and a length that is twice its width. The box will have a volume of 36ft^3. What should the dimensions of the box be in order to minimize the surface area?

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$L e t, B r e a d t h o f t h e b o x = b = x L e n g t h o f t h e b o x = l = 2 b = 2 x (a c c o r d i n g t o q u e s t i o n) H e i g h t o f t h e b o x = h V o l u m e o f t h e b o x = l \times b \times h V o l u m e o f t h e b o x g i v e n = 36 f t^{3} T h e n, l \times b \times h = 36 2 x \times x \times h = 36 2 x^{2} \times h = 36 h = \frac{36}{2 x^{2}} h = \frac{18}{x^{2}} . . . . . . . . . . . . . . . . . . . . . . (1) S u r f a c e a r e a o f t h e s o l i d S = 2 (l b + b h + h l) - l b [s u b t r a c t i n g l b b e c a u s e b o x i s o p e n f r o m t o p] S = 2 (2 x \times x + x \times h + h \times 2 x) - 2 x \times x S = 2 (2 x^{2} + x h + 2 h x) - 2 x^{2} S = 2 (2 x^{2} + 3 h x) - 2 x^{2} S = 4 x^{2} + 6 h x - 2 x^{2} S = 2 x^{2} + 6 h x U \sin g e q u a t i o n (1) - S = 2 x^{2} + 6 (\frac{18}{x^{2}}) x S = 2 x^{2} + \frac{108}{x} . . . . . . . . . . . . . . . (2)$

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