Applied Optimization

Turning a real situation into a surface to climb.

12 problems · about 24 minutes · F-IF.C.7, A-CED.A.2

What this lesson teaches

The student sets up and solves optimisation problems arising from applied contexts.

Naming the variables and writing the objective is the hard half of an applied problem.
A constraint usually lets you eliminate a variable before differentiating.
On a closed region, check interior critical points, the boundary and the corners.

Straightforward practice. Get the method working first.

5 problems

Two numbers sum to 24 with the largest product. What is that product?
Answer 144
Why 144.
Why must the boundary be checked on a closed region?
Answer The extreme may sit on the edge, where the gradient need not be zero.
Why Edges hold extremes that no critical point reveals.
Two numbers sum to 20 with the largest product. What is that product?
Answer 100
Why 10 and 10.
On a closed bounded region, is an absolute maximum guaranteed? 1 yes, 0 no.
Answer 1
Why Closed and bounded is exactly the condition.
A rectangular region has how many corners to check separately?
Answer 4
Why A rectangle.

The same ideas with more to keep track of.

3 problems

An open box with square base side 2 and volume 12. What is its height?
Answer 3
Why 12 divided by 4.
Two numbers sum to 14 with the largest product. What is that product?
Answer 49
Why 7 and 7.
A model gives a side length of −3 metres. Is that a valid answer? 1 yes, 0 no.
Answer 0
Why Lengths cannot be negative.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each location by whether it must be checked when hunting absolute extrema on a closed region.
Answer Must be checked: Interior critical points, The boundary curve, The corners · Not a candidate: Points outside the region
Why Anything outside the region is not allowed at all.
A square base of side 5 with height 4. What is the volume?
Answer 100
Why 25 × 4.
The Box: A box has a square base of side 4 and volume 32. What is its height?
Answer 2
Why 2.
The Two Numbers: Two positive numbers sum to 30 and their product is largest. What is that product?
Answer 225
Why 15 × 15 = 225.