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# Calculus/Maximum-minimum problem

Question
I am struggling with one math questions, and seems to hit the wall. Here is the problem:

A square based box is to be made to have a volume of 15ft^3. The materials for the bottom of box cost 3 times as much per cm^2 as the materials for the top and sides of the box. Find the dimensions for the box that minimize the cost to produce a box.

Here is what I tried:

Volume= Length * Width * Height. Since it is a square box, I assumed that length=Width. Hence, V = x^2 * Height, which equals 15 ft^3. From here, my Height (H) = 15/(x^2).

Now, I need 1 bottom piece, 4 side pieces, and 1 top piece to create a box.
1. Bottom piece = x*x = x^2
2. 4 Side Pieces = 4* (x*h)
3. Top Piece = x*x = x^2

Cost of bottom is 3 times more than sides and top, thus, my surface area = Bottom Piece + 3 *(4 Side Pieces) + 3* Top Piece

Area = x^2 + (12*x*h) + (3*x^2) = 4*x^2 + (12*x*h)

Now representing H in terms of H

Area = 4x^2 + (12*x*(15/x^2)) = 4x^2 + (180/x)

This is where I am stuck now.

Questioner:Daniel
Category:Calculus
Private:No
Subject:Calculus - Optimization / Cost Minimization
Question:

I am struggling with one math questions, and seems to hit the wall. Here is
the problem:

A square based box is to be made to have a volume of 15ft^3. The materials for the bottom of box cost 3 times as much per cm^2 as the materials for the top and sides of the box. Find the dimensions for the box that minimize the cost to produce a box.

Here is what I tried:

..............

Variables:

Let the width and length = x.
Let the height          = h

Then:

Area of base = x^2
Cost of base = 3x^2

Area of top   = x^2
Cost of top   = x^2

Area of one side = hx
Cost of four sides = 4hx

Total C = 4x^2 + 4hx

Now apply your constraint:  hx^2 = 15,  or  h = 15/x^2

You will get

C = 4x^2 + 4(15/x^2)x

.................................
Volume= Length * Width * Height. Since it is a square box, I assumed that length=Width. Hence, V = x^2 * Height, which equals 15 ft^3. From here, my

Height (H) = 15/(x^2).

Now, I need 1 bottom piece, 4 side pieces, and 1 top piece to create a box.
1. Bottom piece = x*x = x^2    <<<<< MULT ONLY THIS BY 3.
2. 4 Side Pieces = 4* (x*h)
3. Top Piece = x*x = x^2

Cost of bottom is 3 times more than sides and top, thus, my surface area =

Bottom Piece + 3 *(4 Side Pieces) + 3* Top Piece

>>> Why this 3? ^     and this one? ^

Area = x^2 + (12*x*h) + (3*x^2) = 4*x^2 + (12*x*h)

Now representing H in terms of H

Area = 4x^2 + (12*x*(15/x^2)) = 4x^2 + (180/x)

This is where I am stuck now.
.....................................
Corrected to:

C = 4(x^2 + 15/x)

The rest is routine -- differentiate, set = 0, solve, etc.

See:

http://en.allexperts.com/q/Calculus-2063/2009/11/Maximum-minimum-problem-41.htm

for LOTS of these examples.

Calculus

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#### Paul Klarreich

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All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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