Section 4.9

More Optimization

From designing minimal-cost packaging to finding the fastest route across different terrains. The math gets messier, but the strategy stays the same.

1

Advanced Strategies

  • Distance Problems

    Tip: Minimize the square of distance instead of . It avoids the messy square root derivative and gives the same location for the minimum.

  • Geometry Constraints

    Often involves Volume vs Surface Area. Use the fixed volume to solve for in terms of .

2

Worked Example: Closest Point

Minimizing Distance

Find the point on the parabola that is closest to the point .

Distance Minimizer

Distance: 4.12

Notice: The shortest path is exactly perpendicular to the curve's tangent line.

Notice the Minimum Distance occurs where the connector is perpendicular to the tangent.

1. Distance Formula

Substitute :

2. Derivative
3. Solve

If , then .
Point is .

3

Level Up Examples

Example A: The Efficient Can

Design a cylindrical can with volume to minimize material (Surface Area).

Cylindrical Can Optimization
1. Constraint (Volume):
2. Primary (Area):

Sub h:

3. Optimize:
.

Example B: Optimal Path

Dog runs at and swims at . Ball is down shore and out. How far down the shore () should it run before jumping in?

Dog Beach Optimization Diagram
1. Time Formula:

Run meters, Swim diagonal to .

2. Derivative:

3. Optimize:

Set :

Squaring both sides and solving yields meters.

Run almost all the way there to maximize the faster land speed!

5

Practice Quiz

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