C3 January 2012 Q3

EdexcelOld spec6 marksLogs & Exponentials

3. The area, \(A\) mm\(^2\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by

\[A = 20\mathrm{e}^{1.5t}, \quad t \geqslant 0\]

(a) Write down the area of the culture at midday. (1)

(b) Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute. (5)

Scheme	Marks
20 (mm\(^2\))	B1
	(1)

Mark scheme (b)

Scheme	Marks
\('40' = 20e^{1.5t} \quad \to \quad e^{1.5t} = c\)	M1
\(e^{1.5t} = \dfrac{40}{20} = (2)\)	A1
Correct order \(1.5t = \ln'2' \to \quad t = \dfrac{\ln c}{1.5}\)	M1
\(t = \dfrac{\ln 2}{1.5} = (\text{awrt } 0.46)\)	A1
12.28 or 28 (minutes)	A1
	(5)
	(6 marks)

Notes

M1 Substitutes A=40 or twice their answer to (a) and proceeds to \(e^{1.5t} = \text{constant}\). Accept non numerical answers.

A1 \(e^{1.5t} = \frac{40}{20}\) or 2

M1 Correct ln work to find t. Eg \(e^{1.5t} = \text{constant} \to 1.5t = \ln(\text{constant}) \to t =\)
The order must be correct. Accept non numerical answers. See below for correct alternatives

A1 Achieves either \(\dfrac{\ln(2)}{1.5}\) or awrt 0.46 2sf

A1 Either 12:28 or 28 (minutes). Cao

Alternatives in (b)

Alt 1- taking ln’s of both sides on line 1

M1 Substitutes A=40, or twice (a) takes ln’s of both sides and proceeds to \(\ln('40') = \ln 20 + \ln e^{1.5t}\)

A1 \(\ln(40) = \ln 20 + 1.5t\)

M1 Make t the subject with correct ln work.
\(\ln('40') - \ln 20 = 1.5t\) or \(\ln\left(\dfrac{'40'}{20}\right) = 1.5t \to t =\)

A1,A1 are the same

Alt 2- trial and improvement-hopefully seen rarely

M1 Substitutes t= 0.46 and t=0.47 into \(20e^{1.5t}\) to obtain A at both values. Must be to at least 2dp but you may accept tighter interval but the interval must span the correct value of 0.46209812

A1 Obtains A(0.46)=39.87 AND A(0.47)=40.47 or equivalent

M1 Substitutes t=0.462 and t=0.4625 into \(40e^{1.5t}\)

A1 Obtains A(0.462)=39.99 AND A(0.4625)=40.02 or equivalent and states t=0.462 (3sf)

A1 AS ABOVE

No working leading to fully correct accurate answer (3sf or better) send/escalate to team leader