Find the general solution to $\frac{dx}{dt}=1+t-x-xt$.

(In each option, $C$ is an arbitrary constant.) Exactly one option must be correct)

(In each option, $C$ is an arbitrary constant.) Exactly one option must be correct)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

Find the general solution to ${x}^{2}\frac{dy}{dx}+2xy={cos}^{2}x$.

(In each option, $C$ is an arbitrary constant.) Exactly one option must be correct)

(In each option, $C$ is an arbitrary constant.) Exactly one option must be correct)

*Choice (a) is correct!*

*Choice (b) is incorrect*

Add the arbitrary constant
before dividing by ${x}^{2}$.

*Choice (c) is incorrect*

*Choice (d) is incorrect*

A cup of coffee has a temperature of
$9{5}^{\circ}$C and is in a room where
the temperature is $2{0}^{\circ}$C.

Let $T=$ the temperature of the coffee after $t$ minutes.

Assuming Newton’s Law of Cooling, which of the following options describes $T$ ? Exactly one option must be correct)

Let $T=$ the temperature of the coffee after $t$ minutes.

Assuming Newton’s Law of Cooling, which of the following options describes $T$ ? Exactly one option must be correct)

*Choice (a) is incorrect*

Note
that $\frac{dT}{dt}$, must be negative,
since the coffee is cooling.

*Choice (b) is correct!*

*Choice (c) is incorrect*

The rate of change of $T$
is proportional to the difference between
$T$ and the temperature
of the surroundings.

*Choice (d) is incorrect*

The rate
of change of $T$ is proportional
to the difference between $T$
and the temperature of the surroundings.

A learning curve is the graph of a function
$P\left(t\right)$, the
performance of someone learning a skill as a function of the training time. Let
$M$ be
the maximum level of performance of which the learner is capable. Suppose that the
rate at which the performance improves is proportional to the difference
between the maximum level and the current level. Write down the differential
equation which models the learning. Exactly one option must be correct)

*Choice (a) is incorrect*

$M$ is a
constant. It is $P$
that is changing.

*Choice (b) is incorrect*

Try again.

*Choice (c) is incorrect*

$\frac{dP}{dt}$ must
be positive.

*Choice (d) is correct!*

Suppose that a tumour in a rat is approximately spherical, and that its rate of
growth is proportional to its diameter. If the tumour has diameter 5 mm when
detected, and 8 mm three months later, what will the diameter be after another three
months?

Give your answer correct to the nearest millimetre.

Give your answer correct to the nearest millimetre.

*Correct!*

*Incorrect.*

*Please try again.*

Water leaks out of a barrel at a rate proportional to the square root of
the depth of the water at the time. If the water level starts at 36 cm and
drops to 35 cm in one hour, how long will it take for all the water to leak
out?

Give your answer in hours, correct to the nearest hour.

Give your answer in hours, correct to the nearest hour.

*Correct!*

*Incorrect.*

*Please try again.*

Start with the differential equation
$\frac{dh}{dt}=k\sqrt{h}$, where
$h$ is the depth of the
water at time $t$.

A tank holds 1000 litres of water, in which 15 kg of salt is dissolved. Pure
water enters the tank at the rate of 10 litres per minute. The solution is
kept thoroughly mixed and is drained from the tank at the same rate. If
$m$ is the mass of salt
in the tank at time $t$,
which of the following options describes the rate of change of the
mass of salt in the tank? Exactly one option must be correct)

*Choice (a) is incorrect*

No salt is being
added to the tank.

*Choice (b) is correct!*

*Choice (c) is incorrect*

The rate at which salt is removed is not constant; it depends on
$m$.

*Choice (d) is incorrect*

Try again. Remember
that $\frac{dm}{dt}=$ rate at which
salt is being added $-$
rate at which it is being removed.

A tank holds 1000 litres of pure water. Brine which contains 0.05 kg of salt per litre
enters the tank at the rate of 5 litres per minute. The solution is kept thoroughly
mixed and is drained from the tank at the rate of 5 litres per minute. If
$m$ is the mass of salt
in the tank at time $t$,
which of the following options describes the rate of change of the
mass of salt in the tank? Exactly one option must be correct)

*Choice (a) is incorrect*

Salt is being added to the tank, as well as being removed.

*Choice (b) is incorrect*

Salt is added at the rate
of $0.25$ kg per minute, but it
is also being removed.

*Choice (c) is correct!*

*Choice (d) is incorrect*

Salt is being added
at the rate of $5\times 0.05$
kg per minute.

For the tank described in Question 8, determine the mass
$m$ of salt in the
tank at time $t$ as
a function of $t$.
(That is, solve the differential equation that is the correct
answer to Question 8.) Exactly one option must be correct)

*Choice (a) is incorrect*

$\int \frac{1}{50-m}\phantom{\rule{0.3em}{0ex}}dm=-ln\left(50-m\right)$.
Also, an initial condition is given, so the value of
$A$ should be
found.

*Choice (b) is incorrect*

$\int \frac{1}{50-m}\phantom{\rule{0.3em}{0ex}}dm=-ln\left(50-m\right)$.

*Choice (c) is incorrect*

An initial condition is given, so the value of
$A$ should be
found.

*Choice (d) is correct!*

A large tank (with capacity 500 litres) contains 100 litres of fresh water. A solution
with a salt concentration of 0.4 kg per litre is added at a rate of 5 litres per minute.
The solution is kept mixed and is drained from the tank at the rate of 3 litres per
minute.

Find the concentration of salt in the tank after $20$ minutes. Give your answer in kg/litre, correct to two decimal places.

Find the concentration of salt in the tank after $20$ minutes. Give your answer in kg/litre, correct to two decimal places.

*Correct!*

Well done!

*Incorrect.*

*Please try again.*

If
$y$ is the amount of salt
(in kg) in the tank after $t$
minutes, the differential equation to be solved is
$\frac{dy}{dt}=2-\frac{3y}{100+2t}$.
Find the particular solution to this equation corresponding to the fact
that there is no salt in the tank initially, and then find the value of
$y$ when
$t=20$.