Exercises - Global Extrema and Optimization; Related Rates
We start with straightforward global extrema problems, then we look at some
optimization and at the end we do some
related rates problems.
If you want to refer to sections of Methods Survey while working the exercises,
you can click
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separate full-size window. Similarly,
here we offer Theory.
In the following problems, find global extrema of the given function over
the given set.
Solve the following optimization problems.
- 10. Find a real number x that minimizes
2x + x2.
Hint
Answer
- 11. Find the x that minimizes
x + 100/x over the set of positive
numbers.
Hint
Answer
- 12. Among rectangles with perimeter 40, which one has the
largest area?
Hint
Answer
- 13. What point on the line
y = 2x + 2 is nearest to the
origin?
Hint
Answer
- 14. What point on the hyperbola
x2/2 − y2 = 1
is nearest to the point (3,0)?
Hint
Answer
- 15. Among rectangles inscribed into a circle of radius R,
which one has the largest area?
Hint
Answer
- 16. Among rectangles inscribed into a half-circle of radius
R, which one has the largest area?
Hint
Answer
- 17. Find which cylinder inscribed into a ball of radius R
has the largest volume.
Hint
Answer
- 18. A corridor 3 m wide turns into a corridor 4 m wide, the two
corridors are perpendicular. What is the longest beam that you can drag on
the floor through this turn? The beam is so thin that you can ignore its
thickness and
just work with a straight segment.
Hint
Answer
Remark: We drag the beam in order to keep the problem two-dimensional. If we
allowed other positions, then obviously the best way to carry it is so that
one ends trails the ground and the other touches the ceiling. Then the length
of the longest beam is given by the length of the beam's projection onto the
floor - and we are back to the 2D problem we ask about here. Thus the problem
that we ask about also pretty much solves the general situation.
If there is no ceiling, then we can take beam of arbitrary length - we
simply carry it in vertical position.
If we had to consider the width of the object - e.g. if the question was
what is the longest sofa 1 m wide one can move through - then the problem becomes
much more difficult, not in idea, but in calculations. If you are curious,
check out
this note.
Solve the following "related rates" problems.
Note: If you feel uneasy about units, just replace meters with feet and km
with miles and everything will work just the same. Can't help you with
radians, getting around them is too much work to be worth it.
- 19. A fisherman is sitting on a bank of a straight river. A fish
takes a bite when it is 2 meters from the fisherman exactly opposite him,
and starts swimming at 5 m/sec along the river. How fast is the
line reeling off?
Hint
Answer
- 20. We have a magical rectangle whose one side can be
shortened/lengthened and the other adjusts automatically so that the area of
the rectangle stays constant at A. At what rate does one side shorten
if we lengthen the other side at the rate k?
Hint
Answer
- 21. You stand at a lamppost that has light 6 m above the ground.
You are 2 meters tall. You start walking away at 1 m/sec.
How fast is the length of your shadow increasing when you are 4 meters from
the lamp?
Hint
Answer
- 22. A lamp lies on the ground 10 meters from a vertical wall. A
man 2 meters tall walks at 1/2 m/sec from the lamp directly
towards this wall. How fast is the height of his shadow changing when he is
halfway to the wall? Is the shade growing or getting shorter?
Hint
Answer
- 23. An airplane flies over a radar site at elevation
5 km, it flies straight at 780 km/hr. What
relative speed will the radar measure when the plane is 13 km away?
Hint
Answer
- 24. A camera positioned on the ground tracks a rocket that took
off from a launching pad that is 1 km from the camera. The rocket flies
straight up with acceleration 10 m/sec2. At what
rate is the camera swiveling up 20 seconds after the take-off?
Hint
Answer
- 25. Consider a triangle with a certain angle a and
adjacent sides of length 4. How fast is its area changing when we start
changing a?
Hint
Answer
- 26. A ladder 6 meters long is lying on the ground perpendicularly
to a vertical wall, with one end right against this wall.
We tie a rope to this adjacent end and start pulling it up along the wall
at 1/2 m/sec. How fast does the other end move along the ground
when the end with the rope is half the way up?
Hint
Answer
- 27. A ladder 6 meters long is lying on the ground perpendicularly
to a vertical wall, with one end right against this wall.
We tie a rope to the opposite end and start pulling it towards the wall
(that is also 6 m tall) at 1/2 m/sec, forcing the
ladder to start turning about the adjacent end. How fast is the angle
between the ladder and the ground changing?
Hint
Answer
- 28. A conveyor belt pours sand on a heap at the rate
0.1 m3/sec. The falling sand forms a regular cone
whose height is always equal to the radius of its base. How fast is the
height of the heap increasing when it is 2 meters tall?
Hint
Answer
- 29. You are on a biking trip with your friend and at a crossing
decide to split. The two roads take you in straight and perpendicular
directions. You bike at 20 km/hr, your friend bikes at
15 km/hr. How fast is your distance growing after 2 hours?
Hint
Answer
- 30. Two roads run parallel and straight 10 km apart.
On each road there is a car, one goes at 50 km/hr, the other at
56 km/hr in the same direction. At time 0 they are level. How
fast is their distance increasing 4 hours later?
Hint
Answer
- 31. A cylinder is rolling down a slope, picking up snow along the
way, thus increasing its radius. How fast is its volume growing depending
on the rate of change of radius?
Hint
Answer
- 32. Picture a wheel powered by a steam engine directly via piston
and rod. The piston moves to and fro in a straight line, at its end there is
a rod 4 meters long whose other end is fixed to the rim of the wheel that has
diameter 1 meter. The center of this wheel is exactly on the axis of the
piston. The wheel turns at a constant speed so that the whole rig moves at
10 m/sec. Find the relative velocity of the piston with respect
to the engine.
Hint
Answer
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