Date | May 2008 | Marks available | 13 | Reference code | 08M.1.hl.TZ1.13 |
Level | HL only | Paper | 1 | Time zone | TZ1 |
Command term | Find and Show that | Question number | 13 | Adapted from | N/A |
Question
A gourmet chef is renowned for her spherical shaped soufflé. Once it is put in the oven, its volume increases at a rate proportional to its radius.
(a) Show that the radius r cm of the soufflé, at time t minutes after it has been put in the oven, satisfies the differential equation drdt=kr, where k is a constant.
(b) Given that the radius of the soufflé is 8 cm when it goes in the oven, and 12 cm when it’s cooked 30 minutes later, find, to the nearest cm, its radius after 15 minutes in the oven.
Markscheme
(a) dVdt=cr A1
V=43πr3
dVdt=4πr2drdt M1A1
⇒4πr2drdt=cr M1
⇒drdt=c4πr A1
=kr AG
[5 marks]
(b) drdt=kr
⇒∫rdr=∫kdt M1
r22=kt+d A1
An attempt to substitute either t = 0, r = 8 or t = 30, r = 12 M1
When t = 0, r = 8
⇒d=32 A1
⇒r22=kt+32
When t = 30, r = 12
⇒1222=30k+32
⇒k=43 A1
∴
When t = 15 , \frac{{{r^2}}}{2} = \frac{4}{3}15 + 32 M1
\Rightarrow {r^2} = 104 A1
r \approx 10{\text{ cm}} A1
Note: Award M0 to incorrect methods using proportionality which give solution r = 10 cm .
[8 marks]
Total [13 marks]
Examiners report
Candidates found this question quite difficult, with only the better students making appreciable progress on part (a). Relatively few candidates recognised that part (b) was asking them to solve a differential equation. Many students tried methods involving direct proportion, which did not lead anywhere.