Length and Girth for Maximum Volume
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A package delivery service expresses its limit on the size of packages as the sum of length and girth must be less than 120 inches.
Referring to the figure, determine the dimensions x and L which result in a package of maximum volume, V.
How do we proceed?
Express the volume in terms of x and L. Use the size limit to relate these two variables to each other. Rewrite the volume in terms of one variable. Differentiate and set to 0.
Express the volume in terms of x and L.
Equation 1
Express the size limit, 120, in terms of x and L.
Equation 2
Solve equation 2 for L.
Equation 3
Substitute equation 3 into equation 1.
We get
Distribute the multiplication.
Equation 4
Take the derivative.
Equation 5
Factor and set to 0.
Solve for the maximizing value of x.
We get x = 0 or 20. We can discard the 0-value because it would lead to a vloume of zero in equation 1.
Are we done?
No.
What else must we do
We need to determine the corresponding values of L and V.
How can we do that?
We can use equations 3 and 1. Do that.
From equation 3,
From equation 1,
How can we check whether this is a maximum or a minimum?
We can determine the second derivative of V and
check the algebraic sign for x = 20.
Do that, using equation 5.
Evaluate for x = 20.
What is our conclusion?
Since the sign is negative, we can conclude that there is a maximum at x = 20.
For further confirmation, graph equation 4 for 0 < x < 25. Then check with the graph below.
Volume versus x, showing the maximum at x = 20.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor