Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 7: Marked Homework Assignment
Due: Thu 2011 Mar 3 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. A dam 200 m long and 24 m high presents a slanted face to the water in the reservoir
behind the dam. The distance from the bottom of the dam to the top along the slanted
face is 26 m. If the surface of the water is level with the top of the dam, find the total
force of water on the dam. The density of water is 1000 kg/m3 .
2. A cubical metal tank 4 ft × 4 ft × 4 ft has one wall in the xy-plane, centred on
the y-axis, with its bottom edge along the x-axis. Near the bottom of this wall is a
parabolic window, in the finite region bounded by the curves y = x2 and y = 1 (x and
y measured in ft), that is designed to withstand a hydrostatic force of 160 lb without
cracking or leaking. The liquid in the tank weighs 50 lb/ft3 . (a) What is the force on
the window when the liquid in the tank is 2 ft deep? (b) What is the maximum height
to which the tank can be filled without exceeding the design limitation?
3. Find the centroid of the plane region bounded by the curves y 2 = x and x − y = 2.
4. Find the centroid of the plane region between y = sin x and y = cos x, with 0 ≤ x ≤
π/4.
5. Solve the initial-value problems:
(a) y 0 = y 2 ,
y(0) = 1.
(b) y 0 = y 2 , y(0) = 0.
√
=
P t, P (1) = 2.
(c) dP
dt
6. A room containing 1000 ft3 of air is originally free of carbon monoxide (CO). Beginning
at time t = 0, cigarette smoke containing 4% CO (by volume) is blown into the room
at the rate of 0.1 ft3 /min, and the well circulated mixture leaves the room at the same
rate. Find the time when the CO concentration in the room reaches 0.012%.