June 2022 Paper 1 Q6

OCR ACurrent spec11 marksLogs & Exponentials Modelling

6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula.

\(P = 100\mathrm{e}^{t}\)

(a) Use this model to answer the following.

(i) Find the value of \(P\) when \(t = 4\). [1]

(ii) Find the value of \(t\) when the population is 9000. [2]

(b) It is suspected that a more appropriate model would be the following formula.

\(P = ka^{t}\) where \(k\) and \(a\) are constants.

(i) Show that, using this model, the graph of \(\log_{10} P\) against \(t\) would be a straight line. [2]

Some observations of \(t\) and \(P\) gave the following results.

\(t\)	1	2	3	4	5
\(P\)	100	500	1800	7000	19000
\(\log_{10} P\)	2.00	2.70	3.26	3.85	4.28

(ii) On the grid below, draw a line of best fit for the data points \((t, \log_{10} P)\) given in the table. [2]

Blank grid with log10 P from 0 to 5 on the vertical axis and t from 0 to 6 on the horizontal axis

(iii) Hence estimate the values of \(k\) and \(a\). [4]

Mark scheme (a)

Scheme	Marks	AO
(i) 5460 (3 sf)	B1	1.1
	[1]
(ii) \(9000 = 100\mathrm{e}^{t}\) \(t = \ln 90\)	M1	3.1a
\(= 4.50\) (3 sf) Allow 4.5 ISW	A1	1.1
	[2]

Notes

(a)(ii)

M1: May be implied by answer

A1: Ignore units. Decimal answer needed

Mark scheme (b)

Scheme	Marks	AO
(i) \(\log_{10} P = \log_{10}(ka^{t})\)
\(\log_{10} P = \log_{10} k + \log_{10}(a^{t})\)	M1	1.1
\(\log_{10} P = \log_{10} k + t\log_{10} a\)	A1	1.1
	[2]
(ii) Points plotted correctly \(\pm 0.1\)	B1	1.1
Line of best fit drawn, between \((1, 2.0)\) and \((1, 2.4)\) and between \((5, 4.2)\) and \((5, 4.5)\)	B1f	1.1
	[2]
(iii) Read off \(c\) and attempt \(10^{c}\). May be implied by value of \(k\)	M1	3.1a
\(k = 19.9\) to \(63.1\)	A1	2.1
Attempt gradient of their graph AND correct ft equation in \(a\). May be implied by value of \(a\)	M1	1.1
\(a = 3.16\) to \(5.01\) (3 sf)	A1	1.1
	[4]

Notes

(b)(i)

First line: No marks yet

M1: At least two terms correct, may be implied by next line

A1: All correct, in this form

(b)(ii)

B1: NB. May be implied by correct line of best fit

B1f: ft reasonable line through their points

(b)(iii)

M1: ft their line. Probably \(c = 1.3\) to \(1.8\), \(k = 10^{1.3}\) to \(10^{1.8}\)

M1: ft their line. Probably \(m = 0.5\) to \(0.7\) AND \(\log_{10} a = 0.5\) to \(0.7\) OR \(a = 10^{0.5}\) to \(10^{0.7}\) scores

NB Use of two points and simultaneous equations: no marks unless the two points used are on their line of best fit.

If first method used for \(k\) or \(a\) and then one point substituted in equation to find the other letter, no marks for second letter unless point used is on line of best fit.