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Quiz 10: Solving Differential Equations; Modelling

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Question 1

Find the general solution to dx-= 1+ t- x - xt dt .
(In each option, C is an arbitrary constant.)

a)
x = Ce -t
  b)
2 x = 1- Cet+t∕2
c)
2 x = 1- Ce -t-t ∕2
  d)
2 2 x = e-t-t ∕2(t+ t2 + C )

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.

Question 2

Find the general solution to x2dy-+ 2xy = cos2x dx .
(In each option, C is an arbitrary constant.)

a)
1 sin2x C y = 2x-+ -4x2-+ x2
  b)
-1- sin-2x- y = 2x + 4x2 + C
c)
sin2-x+-C- y = x2
  d)
x- sin-2x- y = 2 + 4 + C

 

Your answer is correct.
Not correct. Choice (b) is false.
Add the arbitrary constant before dividing by x2.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

A cup of coffee has a temperature of 95∘C and is in a room where the temperature is 20∘C.
Let T = the temperature of the coffee after t minutes.
Assuming Newton’s Law of Cooling, which of the following options describes T ?

a)
dT- dt = k(T - 20), k a positive constant.
   b)
dT- dt = -k(T - 20), k a positive constant.
c)
dT- dt = -k(T - 95), k a positive constant.
   d)
dT dt- = k(T - 95), k a positive constant.

 

Not correct. Choice (a) is false.
Note that dT- dt, must be negative, since the coffee is cooling.
Your answer is correct.
Not correct. Choice (c) is false.
The rate of change of T is proportional to the difference between T and the temperature of the surroundings.
Not correct. Choice (d) is false.
The rate of change of T is proportional to the difference between T and the temperature of the surroundings.

Question 4

A learning curve is the graph of a function P(t), 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.

a)
dM-- dt = k(M -P), k a positive constant.
   b)
dP- dt = kM - P, k a positive constant.
c)
dP- dt = -k(M - P), k a positive constant.
   d)
dP- dt = k(M - P), k a positive constant.

 

Not correct. Choice (a) is false.
M is a constant. It is P that is changing.
Not correct. Choice (b) is false.
Try again.
Not correct. Choice (c) is false.
dP- dt must be positive.
Your answer is correct.

Question 5

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.

 

Your answer is correct
Not correct. You may try again.

Question 6

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.

 

Your answer is correct
Not correct. You may try again.
Start with the differential equation dh- dt = k√h--, where h is the depth of the water at time t.

Question 7

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?

a)
dm- -m- dt = 15- 100
  b)
dm- = - m-- dt 100
c)
dm- = - 15- dt 100
  d)
dm-= 15---m dt 1000

 

Not correct. Choice (a) is false.
No salt is being added to the tank.
Your answer is correct.
Not correct. Choice (c) is false.
The rate at which salt is removed is not constant; it depends on m.
Not correct. Choice (d) is false.
Try again. Remember that dm --- dt = rate at which salt is being added - rate at which it is being removed.

Question 8

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?

a)
dm- = - m-- dt 200
  b)
dm --- = 0.25 dt
c)
dm- 50--m- dt = 200
  d)
dm- = 0.05 - m-- dt 200

 

Not correct. Choice (a) is false.
Salt is being added to the tank, as well as being removed.
Not correct. Choice (b) is false.
Salt is added at the rate of 0.25 kg per minute, but it is also being removed.
Your answer is correct.
Not correct. Choice (d) is false.
Salt is being added at the rate of 5 × 0.05 kg per minute.

Question 9

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.)

a)
m = 50- Aet∕200 , A an arbitrary constant.
   b)
t∕200 m = 50(1- e ) .
c)
m = 50 - Ae-t∕200 , A an arbitrary constant.
   d)
m = 50(1- e-t∕200) .

 

Not correct. Choice (a) is false.
∫ --1--- 50- m dm = - ln(50- m ) . Also, an initial condition is given, so the value of A should be found.
Not correct. Choice (b) is false.
∫ --1--- 50- m dm = - ln(50 - m) .
Not correct. Choice (c) is false.
An initial condition is given, so the value of A should be found.
Your answer is correct.

Question 10

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.

 

Your answer is correct
Well done!
Not correct. You may try again.
If y is the amount of salt (in kg) in the tank after t minutes, the differential equation to be solved is dy dt = 2 ---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.
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