"Published in Suisun City, California, USA"
A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
We know that the volume of a rectangular parallelepiped is
The area of the lower base is equal to
The area of the lateral sides is equal to
but
then the above equation becomes
The total cost of the material is equal to
Take the derivative on both sides of the equation with respect to x, we have
Set dC/dx = 0 because we want to minimize the cost of the material
Therefore, the total cost of the material is
Rationalize the denominator at the above equation, we have
Give the value of the cube roots of the above equation
