92.131
Calculus 1
Optimization Problems
1)
A Norman window has the outline of a semicircle on top of a rectangle as shown in the figure.
Suppose there is
π
+
8
feet of wood trim available for all 4 sides of the rectangle and the
semicircle.
Find the dimensions of the rectangle (and hence the semicircle) that will maximize
the area of the window.
2)
You are building a cylindrical barrel in which to put Dr. Brent so you can float him over Niagara
Falls. I can fit in a barrel with volume equal 1 cubic meter.
The material for the lateral surface
costs $18 per square meter.
The material for the circular ends costs $9 per square meter.
What
are the exact radius and height of the barrel so that cost is minimized?
3)
A rectangular sheet of paper with perimeter 36 cm is to be rolled into a cylinder.
What are the
dimensions of the sheet that give the greatest volume?
4)
A right triangle whose hypotenuse is
3 m long is revolved about one of its legs to generate
a right circular cone.
Find the radius, height, and volume of the cone of greatest volume.
Note:
h
r
V
2
3
1
π
=
.
5)
Determine the cylinder with the largest volume that can be inscribed in a cone of height 8 cm
and base radius 4 cm
.
3
h
r

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