Maximum

Maximum/Minimum Problems

MathsDirect

An open topped box is to be made, with 864cm²of wood. The box is to be twice as long as it is wide. What is the greatest possible volume of the box?

In this case, the first step has to be to draw a diagram of the situation.

Now write down equations for the surface area and volume of the box.

We now need to eliminate h. Remember that in this case we want to maximize the volume, so we make h the subject of the Area equation.


	Substitute h into V and tidy up
	Differentiate V w.r.t. x and say that the result is equal to zero.
	Solve the equation that you have formed, to find the optimal value of x.
	Substitute this value of x back into the Volume equation, to find the greatest possible volume.
	Give your final answer.

Note that throughout the working, the units were omitted, as they would only have cluttered up the equations. It is essential that they return for the final answer. You will be penalized.

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