Question 936346: An artist plans to construct an open box from 15 in. By 20 in. Sheet of met by cutting squares from the corners and folding up the sides. Sketch the graph that will show maximum volume of the box in terms of x.
Define variables (represent length, width and height in terms of x)
Write a polynomial as the product of linear factors
Sketch the graph
Indicate where on the graph the maximum volume will be found
Found 2 solutions by TimothyLamb, josgarithmetic:
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! ---
x = side length of corner squares
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L = 20 - 2x
w = 15 - 2x
h = x
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v = Lwh
v = (20 - 2x)(15 - 2x)x
v = (20*15 - 20*2x - 2*15x + 4xx)x
v = 300x - 40xx - 30xx + 4xxx
v = 4xxx - 70xx + 300x
v = 4x^3 - 70x^2 + 300x
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the graph:
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maximum volume occurs at the graph peak, for x approximately = 2.831 inches
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Answer by josgarithmetic(39620)  (Show Source):
You can put this solution on YOUR website! w=15, width of rectangle
L=20, length of rectangle
x, side length of square removed from the corners
v, volume of rectangular box formed
x is the single variable, and although v is another single variable, v is also the function for volume which depends on x.
The factorized form allows you to find three Real roots for the graph. Checking for signs of v on the intervals is also best done using the factorized form of the function.
Do you know simple derivatives of polynomials? If you do, then you can use this to find maximum and minimum points of the graph. That means, maximum (and minimum) values for the volume v. You might use a graphing calculator or other software to check your work.
The graph is not shown for you in this post, but the code for it would be, unless some graph dimensions need adjustment,
graph(300,300,-4,14,-4,14,(15-2x)(20-2x)x)
(the rendering tags simply not included for the graph code).
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