Lesson 4: Optimization Problems
Estimated time: 40-50 minutes
Learning Objectives
- Set up optimization problems from word descriptions
- Reduce to a single-variable calculus problem using constraints
- Find and verify the optimal solution
Optimization Strategy
Optimization Procedure
- Understand the problem: Draw a picture. Identify what to maximize or minimize.
- Assign variables: Write an equation for the quantity to optimize (the objective function).
- Use the constraint to eliminate a variable so the objective has one variable.
- Find the critical points and determine which gives the optimal value.
- Verify using the First or Second Derivative Test (or compare endpoint values).
Classic Optimization Problems
Example 1: Fencing Problem
A farmer has 200 m of fencing to enclose a rectangular field along a river (no fence needed along the river). What dimensions maximize the area?
Variables: Let x = side perpendicular to river, y = side along river. Constraint: 2x + y = 200, so y = 200 − 2x.
Objective: A = xy = x(200 − 2x) = 200x − 2x².
Optimize: A'(x) = 200 − 4x = 0 ⇒ x = 50. y = 100.
Verify: A''(x) = −4 < 0, confirming a maximum. Max area = 5000 m².
Example 2: Minimum Distance
Find the point on the curve y = √x closest to (3, 0).
Distance: D = √[(x − 3)² + x]. Minimize D² = (x − 3)² + x = x² − 5x + 9.
Optimize: d(D²)/dx = 2x − 5 = 0 ⇒ x = 5/2. y = √(5/2).
Closest point: (5/2, √(5/2)).
Example 3: Open-Top Box
An open-top box is made from a 12 × 12 sheet by cutting squares of side x from each corner. What x maximizes volume?
Dimensions: (12 − 2x) × (12 − 2x) × x, where 0 < x < 6.
Volume: V = x(12 − 2x)² = x(144 − 48x + 4x²) = 4x³ − 48x² + 144x.
Optimize: V' = 12x² − 96x + 144 = 12(x − 2)(x − 6) = 0. x = 2 (since x = 6 is not in domain).
Max volume = 2(8)² = 128 cubic units.
Key Takeaways
- Draw a picture and clearly define the objective function and constraint.
- Use the constraint to reduce to one variable before differentiating.
- Check the domain of your variable and verify your answer is a max/min.
- For a closed domain, compare critical point values with endpoint values.
Check Your Understanding
1. Find two positive numbers whose sum is 100 and whose product is maximum.
2. A cylindrical can must hold 500 cm³. Find the radius that minimizes the surface area.
3. Find the rectangle of maximum area inscribed in a semicircle of radius 5.
4. A box with a square base and open top must have volume 32,000 cm³. What base side minimizes material?