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A rectangular box with a square bottom and closed top is to be made from two materials. The material for the side costs $1.50 per square foot and the material for the top and bottom costs $3.00 per square foot. If you are willing to spend $15 on the box, what is the largest volume it can contain? Justify your answer completely using calculus.

 

If anyone can give step-by-step instructions on how to do this problem, I would really appreciate it.  Thanks in advance.

 Jan 9, 2019
edited by gibsonj338  Jan 10, 2019
 #1
avatar+4457 
+4

\(\text{the wording is kind of odd. Is the bottom open? Sort of a weird way to orient the problem if so}\\ \text{I'm going to assume both the top and bottom are closed}\\ \text{we've got the two different types of materials so let's compute the two areas separately}\\ \text{assume the box has square side length }s \text{ and height }h\\ area_{side}=4sh\\ area_{tb} = 2s^2\\ vol = s^2h\)

 

\(\text{we can convert areas into cost using the materials prices}\\ C = 4sh\cdot (1.5) + 2s^2 \cdot (3) = 6sh + 6s^2\)

 

\(\text{I"m going to assume you haven't been taught Lagrange multipliers yet}\\ \text{so we have to use the cost function to write the volume in a single variable}\\ 15 = 6sh + 6s^2\\ h = \dfrac{15-6s^2}{6s}\\ vol = s^2 h = s^2\dfrac{15-6s^2}{6s} = \dfrac{15s-6s^3}{6}\)

 

\(\text{Now to find the extrema of the volume we set it's derivative equal to 0}\\ \dfrac{dvol}{ds} = \dfrac{15-18s^2}{6} =\dfrac{ 5-6s^2}{2} = 0\\ 5 = 6s^2\\ s = \sqrt{\dfrac 5 6}~ft \text{ (we can ignore the negative solution)}\)

 

\(vol = \dfrac 1 6 \left(15\sqrt{\dfrac 5 6} -6 \left(\dfrac 5 6\right)^{3/2}\right) = \dfrac 5 3 \sqrt{\dfrac 5 6}~ft^3\)

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 Jan 10, 2019
 #3
avatar+17332 
+2

The top is closed per the question....but it does not say if it is the same material as the bottom.....just states BOTTOM is 3/sqft.....

    I'm glad you got an answer!   I got stuck when I tried top made of same material as sides and only bottom at $3/sqft.    ~EP

ElectricPavlov  Jan 10, 2019
 #2
avatar+4457 
+2

\(\text{What's that? Lagrange multipliers?}\\ \text{The method of Lagrange multipliers involves solving}\\ \nabla (f(s,h) - \lambda (c(s,h)-C))\\ \text{In this problem }\\ f(s,h) = vol(s,h) = s^2 h\\ c(s,h)=6sh+6s^2 (\text{see the previous answer})\\ C=15\)

 

\(\text{our Lagrange equation becomes}\\ \nabla \left(s^2 h - \lambda (6sh-6s^2 - 15)\right)= (0,0,0)\\ \text{this leads to the system of equations }\\ 2sh - \lambda(6h - 12s)=0\\ s^2 - 6s\lambda = 0\\ 6sh-6s^2-15 = 0\\\)

 

\(\text{and these are solved for }s,~h,~\lambda\\ \text{This seems like a lot more work than the previous answer. Why do it?}\\ \text{In most problems you can't cleanly solve for one in terms of the other and substitute like we did}\)

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 Jan 10, 2019
 #4
avatar+98154 
+1

Let the volume be

 

V =  x^2*h      where x is the side of the base and h is the height    (1)

 

And.....using the total cost for the surface area....we know that

 

Total cost for top/ bottom + Total cost for sides =  15

 

2x^2(3) + 4xh(1.5) =  15

 

15 - 6x^2  = 6xh

 

[ 15 - 6x^2] / [6x]  = h      (2)

 

 

Put (2)  into (1)  and we have that

 

V = x^2 [ 15 - 6x^2] / [ 6x]

 

V =  x [ 15 - 6x^2] / 6

 

V =  (1/6) (15x - 6x^3 )       take the derivative and set to 0

 

(1/6) (15 - 18x^2) = 0

 

18x^2 =  15

 

x^2  =  15/18    =  5/6

 

x = √[5/6]  

 

So.....the volume of the box is :    (1/6)√[5/6] (15 -6 (5/6) )  = (1/6)√[5/6] (10) =  (5/3)√[5/6] ft^3    

 

 

 

cool cool cool

 Jan 10, 2019

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