1. Angelo, who is in a rowboat 2 kilometers from a straight shoreline wishes to go back to his house, which is on the shoreline and 6 kilometers from the point on the shoreline nearest Angelo. If he can row at 6 kph and run at 10 kph, how should he proceed in order to get to his house in the least amount of time?

Problem Set in Optimization:
1. Angelo, who is in a rowboat 2 kilometers from a straight shoreline wishes to go back to his house,
which is on the shoreline and 6 kilometers from the point on the shoreline nearest Angelo. If he can row
at 6 kph and run at 10 kph, how should he proceed in order to get to his house in the least amount of
time?
2. A closed box with a square base is to have a volume of 2000 cm3
. The material for the top and bottom
of the box costs P30 per square centimeter and the material for the sides cost P15 per square
centimeter. Find the dimensions of the box so that the total cost of material is the least possible and all
its dimensions do not exceed 20 cm.
3. Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a
right circular cone with radius 4 cm and height 3 cm.
4. An offshore oil well located at a point W that is 5 km from the closest point A on a straight shoreline.
Oil is to be piped from W to a shore point B that is 8 km from A by piping it on a straight line underwater
from W to some shore point P between A and B and then on to B via pipe along the shoreline. If the cost
of laying pipe is P10,000,000/km underwater and P5,000,000/km over land, where should the point P be
located to minimize the cost of laying the pipe?
5. A closed cylindrical can is to hold 1 cubic meter of liquid. Assuming there is no waste or overlap, how
should we choose the height and radius to minimize the amount of material needed to manufacture the
can? (Assume that both dimensions do not exceed 1 meter.)
6. A manufacturer estimates that the cost of production of “x” units of a certain item is
C=40x-0.02x2
-600. How many units should be produced for minimum cost?
7. Two posts, one 8 ft. high and the other 12 ft. high, stand 15 ft. apart from each other. They are to be
stayed by wires attached to a single stake at ground level, the wires running to the tops of the posts.
How far from the shorter post should the stake be placed to use the least amount of wire?