optimization

OptimizationAP Calculus

Design a cylindrical container with a volume of 10 ft3 using the least amount of material.

Volume of a cylinder: V = π r2 h Surface Area of a cylinder: A = 2 π r2 + 2 π r

h

This is an example of an optimization problem

Cylinder Problem

Write h in terms of r: 10 = π r2h, h = 10/π r2

Re-write the S.A. formula: A = 2 πr2 + 20/r Find the domain of r: D = (0, ∞) Find A’(r): A’ (r) = 4 πr – 20 r -2

Set A’(r) = 0 for critical points: Find h:

The dimensions for the container are Radius = 1.17 ft, height = 2.34 ft

Cylinder Problem

Cylinder Problem

Find the dimensions of the largest rectangle that can be inscribed in a circle.

Rectangle Inscribed in a Circle

Rectangle Inscribed in a Circle

Circle: x2 + y2 = r2

y = √(r2 – x2)

Rectangle: A=lw =(2x)(2y)A(x) = 4x √(r2 – x2)

2x

2y

Cowboy Clint wants to build a dirt road from his ranch to the highway so he can drive to town to see his girlfriend.

The perpendicular distance from the ranch to the highway is 4 miles.

The town is located 9 miles from that point.

Cowboy Clint’s Problem

4 mi

9 mi

Ranch

Town

Where should Cowboy Clint build his road if the speed limit on the dirt road is 20 mph and the speed limit on the highway is 55 mph?

Cowboy Clint’s Problem

4 mi

9 mi

Ranch

Townx 9-x

Cowboy Clint’s Problem

4 mi

9 mi

Ranch

Town x 9-x

The rate (mph) on each road is constant so D = rt

For the dirt road :

For the highway:

Total Time:

Set T’(x) = 0 to minimize the time.

To minimize the time, check the endpoints and the critical point(s).

Endpoints [0, 9] T(0) = 0.36h, T(1.56) = 0.35h, T(9) = 0.49h

He travel time for Cowboy Clint is minimized if the road joins the highway at a distance of 1.56 miles.

Cowboy Clint’s Problem

4 mi

9 mi

Ranch

Town1.56 7.44

A fertilizer made from N pounds of nitrogen and P pounds of phosphate is used on Kansas farmland, then the yield of corn is

Cost for nitrogen is 25 cents/pound

Cost for phosphate is 20 cents/pound

Agricultural Optimization

If the farmer intends to spend $30 per acre for fertilizer, what combination of nitrogen and phosphate will produce the highest yield of corn?

Cost equation: 0.25N + 0.2P = 30 Solved for P: P = 150 – 1.25N

Agricultural Optimization

The formula as a function of N

Set the derivative equal to 0.

150 – 2.5N = 0, N = 60

B’(N) is undefined at N = 0 and N = 120

Agricultural Optimization

B(0) = 8 B(60) = 28 B(120) = 8

The maximum yield occurs when N = 60 and P = 75.

Agricultural Optimization

Agricultural Optimization