The production of oil P, in barrels per day, from an oil field satisfies the differential equationdPdt=10002+twhere t is measured in days from the start of production.The production of oil att=0is20,000 barrels per day.

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$1252.7\dots \approx 1250$(barrels per day) A1

[1 mark]

This is the increase (change) in $P$ (production per day) between$t=0$and$t=5$(or during the first $5$ days) A1

[1 mark]

METHOD 1

$P=1000 \ln \left(2+t\right)+c$ (M1)A1

$c=20000-1000 \ln 2\approx 19306.8\dots$ (M1)A1

$P=1000 \ln \left(2+t\right)+19300$

METHOD 2

$\underset{20000}{\overset{P}{\int }}\text{d}P=\underset{0}{\overset{t}{\int }}\frac{1000}{2+x}\text{d}x$ (M1)

${\left.P\right]}_{20000}^P=1000{\left.\ln \left(2+x\right)\right]}_0^t$ A1

Note: A1 is for the correct integral, with the correct limits.

$P-20000=1000\left(\ln \left(2+t\right)-\ln 2\right)$ (M1)A1

$P=1000 \ln \left(\frac{2+t}{2}\right)+20000$

[4 marks]