1. AP Calculus

2.  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

3.  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

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

6. Circle: x2 + y2 = r2
y = √(r2 – x2)
Rectangle: A=lw =(2x)(2y)
A(x) = 4x √(r2 – x2)
2y
2x

7.  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.
Ranch
4 mi
Town
9 mi

8.  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?
Ranch
4 mi
x 9-x Town
9 mi

9. The rate (mph) on each road is constant so D = rt
For the dirt road :
For the highway:
Total Time:
Ranch
4 mi
x 9-x Town
9 mi

10.  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

11.  He travel time for Cowboy Clint is minimized
if the road joins the highway at a distance of
1.56 miles.
Ranch
4 mi
1.56 7.44 Town
9 mi

12.  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

13.  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

14.  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

15.  B(0) = 8
 B(60) = 28
 B(120) = 8
 The maximum yield occurs when N = 60 and
P = 75.