Student number
Name [SURNAME(S), Givenname(s)]
MATH 100, Section 110 (CSP)
Week 6: Marked Homework Assignment
Due: Thu 2010 Oct 21 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. A stone is dropped into a lake, creating a circular ripple that travels outward at a
speed of 50 cm/s. Find the rate at which the area within the circle is increasing, after
(a) 1 s, (b) 3 s, (c) 5 s.
2. Let b denote the relative birth rate for a population, the number of births per unit
time per average individual in the population. For example, if a population of 1000
individuals has an instantaneous birth rate of 25 births per year, the birth rate is B =
25
= 0.025 yr−1 (of course this
25 individuals·yr−1 and the relative birth rate is b = 1000
doesn’t mean that each individual literally has exactly 0.025 offspring in one year,
it’s just the instantaneous rate, averaged over the entire population). Similarly, let d
denote the relative death rate for the population. Suppose for a certain population the
relative birth rate is a constant b = 0.025 yr−1 and the relative death rate is a constant
d = 0.016 yr−1 . Suppose at t = 0, the population size is 4200. Find (a) the birth rate
B at t = 0, (b) the death rate D at t = 0, and (c) the growth rate of the population
dN/dt = B − D at t = 0. (d) Determine how long it takes for the population to
quadruple, assuming that the growth rate is proportional to the population size.
3. Uranium-238 has a half-life of 4.5 × 109 years. (a) What percentage of the amount
present now will be remaining after 109 years?
(b) How long will it take for one quarter of the initial amount to decay?
4. A turkey is put into an oven that has a constant temperature of 200o C. A thermometer
embedded in the turkey registers its temperature. When the turkey is first put in the
oven, the thermometer reads 20o C, and 30 minutes later it reads 30o C. If the turkey
is ready to eat when the thermometer reads 80o C, how long will the turkey have to
be in the oven before it is ready to eat? Use Newton’s law of cooling (or warming)
to write down a differential equation for the temperature of the turkey at time t,
then solve this equation to determine when the turkey will be ready to eat.
5. Solve the initial value problem
dy
= ay + b,
dt
y(0) = c,
where a, b and c are given constants, and a 6= 0. (Hint. ay + b = a(y + (b/a)). Consider
letting u(t) = y(t) + (b/a).) Then solve the initial value problem
dy
= −3y + 10,
dt
y(0) = −99.