Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 4: Marked Homework Assignment
Due: Thu 2011 Feb 3 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Suppose an electric charge q1 = 1.0 µC is held fixed, and another electric charge
q2 = −1.5 µC is moved from a point A that is 2.0 cm from q1 , to a point B that is 6.0
cm from q1 . If the force exerted by q1 on q2 is given by Coulomb’s law F (r) = kq1 q2 /r 2 ,
where r is the distance in m between q1 and q2 and k = 9.0 × 109 N · m2 /C 2 , find the
work done, moving the charge q2 from A to B.
2. Water (with density 1000 kg/m3 ) fills a tank in the shape of an inverted right circular
cone with top radius 3 m and depth 4 m. Determine how much work (in J) must be
done to pump all the water out of the tank over the top edge of the tank. (A mass m,
in kg, has weight mg, in N, where g is the acceleration due to gravity. Use the value
g = 9.8 m/s2 .)
3. Water is being drawn from a well 100 ft deep, using a bucket that scoops up 100 lb of
water. The bucket is pulled up at the rate of 2 ft/s, but it has a hole in the bottom
through which water leaks out at a rate of 12 lb/s. Determine how much work is done
in pulling the bucket to the top of the well. Neglect the weight of the bucket, the
weight of the rope, and any work done in overcoming friction. Hint: First consider
the infinitesimal amount of work dW done to raise the bucket an infinitesimally small
distance. During the infinitesimally small time it takes to do this, the amount of water
in the bucket is constant.
4. A leaky bucket of mass m kg is raised from the ground to a height of h m, at a constant
speed with a rope that has a linear density of ρ kg/m. Initially the bucket contains
M kg of water, but the water leaks out of the bucket at a constant rate and finishes
draining just as the bucket reaches the height of h m. Write down a definite integral
giving the work done raising the leaky bucket up to h m, then evaluate the integral.
5. Suppose a microbe population is increasing at a rate of 2000 + t3 individuals per hour,
where t is in hours. If the population at t = 2 is 10,000, determine the average
population over the time interval [2,4].
6. Evaluate the following integrals (show your work):
(a)
(b)
(c)
R1
R
R
0
xe−2x dx.
e−θ cos 2θ dθ.
t sec t tan t dt.
√
(d) 4 ln s ds.
R
√
(e) sin x dx.
R9