A company plans to extract oil from an oil field - Edexcel - A-Level Maths Pure - Question 8 - 2017 - Paper 1

A limitation of using model A is that the linear model suggests that the volume of oil extracted could potentially become negative when $t$ exceeds 9.14 years, which is not realistic, as it is impossible to extract a negative volume of oil.

A suitable exponential model can be expressed as:

$V = A e^{kt}$

Where:

• At $t=0$, the volume is $V(0) = 16000$: $16000 = A e^{0} \Rightarrow A = 16000.$

• At $t=4$, the volume is $V(4) = 9000$: $9000 = 16000 e^{4k} \Rightarrow \frac{9}{16} = e^{4k} \Rightarrow 4k = \ln\left(\frac{9}{16}\right) \Rightarrow k = \frac{1}{4} \ln\left(\frac{9}{16}\right)$

Thus, the function for model B can be given as: $V = 16000 e^{\left(\frac{1}{4} \ln\left(\frac{9}{16}\right)\right)t}$

Using the equation derived for model B:

$V = 16000 e^{\left(\frac{1}{4} \ln\left(\frac{9}{16}\right)\right)t}$

Substituting $t = 3$:

1. Calculate the value: $V(3) = 16000 e^{\left(\frac{1}{4} \ln\left(\frac{9}{16}\right)\right)(3)} \approx 15806.40 \text{ barrels}$

This serves as an estimate for the daily volume of oil extracted after 3 years.