Optimization Project (PART 1)

AP Calculus AB (Vahsen)
Name: ________________________________
Optimization Project (PART 1)
Part 1 of this project (this paper) is due at the end of class on December 15/16. Part 2 of this
project will be completed in class on December 15/16.
Directions for Part 1:
 You must complete FIVE of the following optimization problems. These problems will be worth
half of your project grade (50 points – 10 points each).
 Numerical answers to these problems will be posted online and in rooms 224 and 218.
 Make sure to include units (where appropriate) and to JUSTIFY YOUR ANSWER.
 You may complete the bonus problem for 5 points of extra credit (it will not count as one of
your graded problems).
1. Find the point on the graph of the function f x   x that is closest to the point (4, 0).
2. A rancher has 200 feet of fencing with which to enclose to adjacent rectangular corrals. What
dimensions should be used to that the enclosed area will be a maximum?
AP Calculus AB (Vahsen)
Name: ________________________________
3. A tank with a rectangular base and rectangular sides is to be open at the top. It is to be
constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank
costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost
of the least expensive tank?
4. A Norman window is constructed by adjoining a semicircle to the top of an ordinary
rectangular window. Find the dimensions of a Norman window of maximum area if the total
perimeter is 16 feet.
5. A rectangle is bounded by the x-axis and the semicircle y  9  x 2 . What length and width
should the rectangle have to that its area is a maximum?
AP Calculus AB (Vahsen)
Name: ________________________________
6. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r.
7. Determine the dimensions of a rectangular solid (with a square base) with a maximum volume
if its surface area is 337.5 square centimeters.
8. A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by
cutting out a square from each of the four corners and bending up the sides. Find the largest
volume that such a box can have.
AP Calculus AB (Vahsen)
Name: ________________________________
9. Find the points on the hyperbola y2 – x2 = 4 that are closest to the point (2, 0).
10. A rectangular package to be sent by a postal service can have a maximum combined length and
girth (perimeter of a square cross section) of 108 inches. Find the dimensions of the package of
a maximum volume that can be sent.
BONUS (DO ON SEPARATE PAPER): A pipe is being carried (so that it is parallel to the floor) around a
corner from a hallway that is 8 feet wide to a hallway that is 4 feet wide. What is the maximum length
of the pipe that can be carried around the corner without getting stuck?