Skip to content

Course home

3.6 Integration (A-level only)

3.6 Integration (A-level only)

EasyMediumHard
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288289290291
Question 238
i.

A spherical drop of industrial lubricant is being injected into a precision-engineered cavity. The volume, VVV, of the drop is increasing at a constant rate of 180π mm3 s−1180\pi \text{ mm}^3\text{ s}^{-1}180π mm3 s−1. Calculate the rate of increase of the radius, rrr, of the drop in mm s−1 \text{mm s}^{-1}mm s−1 at the moment when the radius is exactly 3 mm3 \text{ mm}3 mm. [The volume VVV of a sphere of radius rrr is given by V=43πr3V = \frac{4}{3}\pi r^3V=34​πr3]

[5]
ii.

The depth of sediment, y metresy \text{ metres}y metres, settling at the bottom of an industrial filtration tank is monitored. The rate of change of the depth of the sediment is modeled by the differential equation

dydt=ky2 \frac{\text{d}y}{\text{d}t} = \frac{k}{y^2} dtdy​=y2k​

where kkk is a positive constant and ttt hours is the time after monitoring began. Given that:

  • at the start of monitoring (t=0t = 0t=0), the sediment depth was 2 metres2 \text{ metres}2 metres.
  • after 555 hours of monitoring, the sediment depth had reached 4 metres4 \text{ metres}4 metres.
  • after TTT hours of monitoring, the sediment depth reached 6 metres6 \text{ metres}6 metres.

Solve the differential equation to determine the value of TTT. Give your answer to one decimal place.

[8]
Markscheme

3.6 Integration (A-level only) Questions

  1. A Level
  2. /Maths
  3. /3.6 Integration (A-level only)

363 exam-style questions on CCEA A Level Maths 3.6 Integration (A-level only), covering 3.6.1 Integration (A-level only), 3.6.2 Integration (A-level only), 3.6.3 Integration (A-level only), 3.6.4 Integration (A-level only), 3.6.5 Integration (A-level only), 3.6.6 Integration (A-level only), and 3.6.7 Integration (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

Question bank