The given constraints of the value of the surface area
determines the values of <em>x</em> and <em>h</em> at the maximum volume.
Correct response:
- The function to be used to maximize the volume of the open box is;
- At the maximum volume, ,
<h3>Which is the method used to find the maximum volume of the box</h3>
The given parameters are;
Length of the side of the square base of the box = x
Height of the box = h
Surface area of the box = 20
Required:
Function that can be used to maximize the volume.
Solution:
The function for the surface area is; S.A. = 4·h·x + x²
Volume of the box, V = x²·h
The given surface area, S.A. = 20
Therefore;
S.A. = 20 = 4·h·x + x²
Which gives;
The volume as a function of <em>x</em> is given as follows;
At the maximum volume, we have;
x² = 10
x = √10
f''(√10) = -√10 < 0, therefore;
The maximum value of the volume is given at <u><em>x</em></u><u> = √10</u>
Therefore;
At the maximum x = √(10), h =
The values of <em>x</em> and <em>h</em> at the maximum volume are;
- and
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