Maths Quest Maths C Year 12 for QueenslandChapter 5 Differential equations WorkSHEET 5.21
WorkSHEET 5.2Differential equationsName: ______
1 / Find the general solution to . /2 / Find the particular solution to . /
3 / (a)Find the particular solution of
(x) = if f(0) = .
(b)State the largest domain for which the solution applies. / (a)
Therefore, x = Sin–1y + c
where –1 < y < 1
f(0) =
therefore 0 = Sin–1 + c = + c
(b) and since
that is,
and the largest domain is / 6
4 / (a)Find the general solution of
(b)State the largest domain for which the solution applies. / (a)
(b)y exists when / 5
5 / The number of bacterial colonies in a piece of meat on a kitchen bench on a summer day is increasing at a rate proportional to the number present at any minute. If the number of colonies increases by 50% in 100 minutes, how much longer will it be until the number of colonies is double the initial number? Answer to the nearest minute.
Let A = number of bacterial colonies.
Let x = number of minutes,
Let A0 = number of bacterial colonies at x = 0 /
For double the initial number, A = 2A0
x = ln
It will take 71 minutes longer to double the initial number than to increase the initial number by 50%. / 6
6 / If a freezer is at a constant –6ºC and a tray of water at 12ºC is placed in the freezer, its temperature drops by 4ºC in 1 minute. Let T be the temperature and t the time in minutes. Find:
(a)the time taken for the tray of water to reach freezing point 0ºC (to the nearest second)
(b)the temperature of the tray at any time t (and hence find the temperature after 6minutes). /
/ 8
7 / If a freezer is at a constant –6ºC and a bottle of champagne at 24ºC is placed in the freezer, its temperature drops by 4ºC in 1 minute.
(a)Find the time taken (to the nearest second) for the champagne to reach
10ºC, the optimal drinking temperature.
(b)Find the time at which the bottle would explode under ice pressure assuming that it does this at 4ºC.
(c)What is the time range between optimal drinking time and the time at which the champagne reaches exploding temperature? / Cooling Law. T: temperature, t: time (minutes)
(a)When T = 10
(b)
When T = 4,
(c)Time range: t = 4.39 min to 7.68 min
is 460.6 – 263.6 = 197 s / 8
8 / A ‘celery being’, whose core sodium concentration is 37 mg/L, falls from its celery ship into the Sauerkraut Sea of sodium concentration 73 mg/L. The sodium concentration of the ‘celery being’ rises to 38mg/L in 5 minutes. It will not survive if its sodium concentration rises above 38 mg/L.
(a)Assuming that applies, for how long can the ‘celery being’ survive in the sea after it first falls in?
Two minutes after the ‘celery being’ falls in, the celery ship is 200 metres away and a rescue boat is despatched. Due to the salty fog the boat travels to the ‘celery being’ according to the differential equation where x is the distance travelled by the boat towards the ‘celery being’ in metres and t is the time in seconds since the ‘celery being’ fell in.
(b)Express x as a function of t.
(c)How long has the ‘celery being’ been in the water when the rescue boat arrives?
(d)Decide whether the ‘celery being’ is still alive when the rescue boat arrives. / S: sodium concentration, t: time (seconds)
(a)5 minutes or 300 s, as given in the question.
(c)Find t when x = 200.
(d)The ‘celery being’ can survive only 5minutes, so it is dead by the time the rescue boat arrives. / 10
9 / The number of boxes of fish fingers sold per day (N) at a local supermarket in Norway varies with the number of degrees Celsius (T) that the daily maximum temperature is above 5ºC.
No boxes are sold when the daily maximum temperature is 5ºC
(a)Express N as a function of T
(b)Find the number of boxes of fish fingers sold when the maximum temperature is 30ºC
(c)Find the maximum temperature if 280 boxes of fish fingers are sold in one day.
(d)On a particular day the supermarket has a maximum of 550 boxes of fish fingers in stock. What is the maximum temperature for which the supermarket will sell all its fish fingers? / (a)
(b)When T = 25,
(c)When N = 280,
(d)When N = 550,
/ 10
10 / The rate at which a particular drug is absorbed by the body is proportional to the amount of the drug present (D) at any time t. If 50 mL is initially administered to a patient and 50% remains after 4 hours, find
(a)D as a function of t
(b)the amount present after 6 hours.
(c)If the level of the drug in the body is below 10 mL, current test procedures do not detect the presence of this drug. How long after the initial dose will the drug be undetectable?
(d)A patient is given this drug 9.5 hours before competing in a short-distance Olympic race. The test is upgraded so that within this period the test will be positive if the drug can be detected at a level of 7.5 mL. Does this athlete test positive with the upgraded test? Assume that the test takes place exactly half an hour after the start of the race. / D = no. of mL of drug present at time t
(t in hours). 50% remains after 4 hours
(a)D0 = 50 mL = amount of drug in body
(t = 0)
(b)When t = 6,
(c)When D = 10,
(d)The test takes place 10 hours after first taking the drug.
When t = 10,
The athlete tests positive. / 10