Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 7: Marked Homework Assignment
Due: Thu 2010 Oct 28 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. A streetlight is on top of a vertical pole that is 8 m high. A ball is dropped from a
helicopter and falls vertically, landing 3 m away from the base of the streetlight pole.
When the ball is below the level of the streetlight, the light casts a shadow of the ball
on the ground. When the ball is 4 m above the ground, it is falling at a rate of 20
m·s−1 . (a) How fast is the shadow moving when the ball is 4 m above the ground?
(Assume that the ground is level and flat.) (b) When the ball is 4 m above the ground,
how fast is the distance between the ball and its shadow changing?
2. Water is leaking out of an inverted conical tank at a rate of 10,000 cm3 /min at the
same time that water is being pumped into the tank at a constant rate. The tank has
a height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of
20 cm/min when the height of the water is 2 m, find the rate at which water is being
pumped into the tank.
3. A solid metal cone of radius 3 cm and height 4 cm is lowered point first at a rate of 1
cm/s into a tall cylindrical container of radius 5 cm that is partially filled with water
to a depth of 7 cm before the cone is lowered. How fast is the water level rising at the
moment the cone is completely submerged? (Assume the container is tall enough so
that it does not overflow.)
4. Use linear approximation or differentials to estimate (a) ln 0.98, (b) tan 46o .
5. When blood flows along a blood vessel, the flux F (the volume of blood per unit time
that flows past a given point) is proportional to the fourth power of the radius r of the
blood vessel: F = kr 4 , where k is a positive constant. A partially clogged artery can
be expanded by an operation called angioplasty, in which a balloon-tipped catheter is
inflated inside the artery in order to widen it. Use linear approximation or differentials
to estimate how a 5% increase in the radius would affect the flux of blood. Express
your answer as a percentage change (specify increase or decrease) in the flux of blood.
6. The equation y 4 + y 3 + xy = 3 defines y implicitly as a function of x near the point
(1, 1). Use (a) linear approximation, and (b) quadratic approximation at the given
point to estimate the value of y when x = 0.9.
7. Find the quadratic
approximation (or 2nd-degree Taylor polynomial) Q(x) = T2 (x),
√
for f (x) = 1/ x at 9.
8. Find the 8th-degree Maclaurin polynomial T8 (x), for f (x) = cos x.