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Calculus IApplications

Optimization, find the best possible choice.

Turn a word problem into one function. Use its derivative, the allowed choices and the endpoints to find the greatest area, the most volume or the lowest cost.

Build a model · Step 1 of 7

What can change, and what stays fixed?

River · no fence here28 m66
Optimization means finding the largest or smallest possible result. Imagine 40 m of fence beside a straight river. Only three sides need fencing. Name each short side x{x} and the side parallel to the river y{y}. The total fence stays fixed as the shape changes.
Try it: Move the slider. Making the field deeper uses fence that could have made it longer. Predict which shape encloses the most space.
Depth x (metres)6 m
2x+y=40{\displaystyle 2x} + {\displaystyle y} = {\displaystyle 40}

This is the constraint: the rule every allowed shape must obey. At either end of the slider, the rectangle collapses.

Check yourself
Which sides use the 40 m of fence?

Need a hand setting up an optimization problem?

Bring your practice questions. We work through them together until every type feels routine.

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