A scientist is studying the number of bees and the number of wasps on an island - Edexcel - A-Level Maths Pure - Question 13 - 2022 - Paper 1

To find the rate of increase of bees at $t = 10$, we first differentiate the bee model:

$\frac{dN_b}{dt} = 220 \cdot 0.6 e^{0.6t}$

Substituting $t = 10$ gives:

$\frac{dN_b}{dt} = 220 \cdot 0.6 e^{0.6 \times 10} = 132 e^{6}$

Calculating $e^{6} \approx 403.4288$, we have:

$\frac{dN_b}{dt} \approx 132 \cdot 403.4288 \approx 53281.1$

Since this is in thousands, the rate is approximately:

$\frac{53281.1}{1000} \approx 18$

Thus, the number of bees was increasing at a rate of approximately 18 thousand per year.

To find $T$ where the number of bees equals the number of wasps:

Set the equations equal to each other:

$45 + 220 e^{0.6T} = 10 + 800 e^{0.6T}$

Rearranging gives:

$220 e^{0.6T} - 800 e^{0.6T} = 10 - 45$

This simplifies to:

$$e^{0.6T} = \frac{35}{580} = \frac{7}{116}$$ Taking the natural logarithm of both sides: $$0.6T = \ln\left(\frac{7}{116}\right)$$ Thus: $$T = \frac{\ln\left(\frac{7}{116}\right)}{0.6}$$ Calculating this gives: $$T \approx 12.08$$ Therefore, the value of $T$ is approximately 12.08.