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?


4 mi

9 mi

Ranch

Townx 9-x


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.


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


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


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

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


optimization

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