5. (a) A model for the population, P, of elephants in Serengeti National Park is $$\displaystyle P = \frac{21000}{7 + 3e^{-\frac{t}{3}}}$$ where t is the time in years from today - HSC - SSCE Mathematics Extension 2 - Question 5 - 2008 - Paper 1

To find the population today, substitute t = 0 into the population model:

$P(0) = \frac{21000}{7 + 3e^0} = \frac{21000}{10} = 2100.$

Thus, the population today is 2100 elephants.

As t approaches infinity, e^{-\frac{t}{3}} approaches 0, hence:

$P = \frac{21000}{7 + 0} = 3000.$

Thus, the eventual population predicted by the model is 3000 elephants.

To find the growth rate today, we calculate:

$\text{Annual Growth Rate} = \frac{dP}{dt} \bigg|_{t=0} = \frac{7000}{10^2} = 70.$

Thus, the annual percentage rate of growth is:

$\frac{70}{2100} \times 100 \approx 3.33\%.$

Therefore, the annual percentage rate of growth today is approximately 3.33%.

To show that p(n) has a double zero at x = 1, we compute:

1. Substitute x = 1 into p(n): $p(1) = 1^n + (n + 1)(1) + n = 1 + n + 1 + n = 2n + 2$ which equals 0 when n = -1.

2. Differentiate p(n): $p'(n) = n x^{n-1} + (n + 1)$ Substituting x = 1 gives $p'(1) = n + n + 1 = 2n + 1.$ This equals zero when n = -1, indicating a double zero at x=1.

To show that p(x) ≥ 0 for x ≥ 0, we analyze:

1. Considering concavity, the second derivative test can confirm concavity and positivity.

2. We find that p(x) is non-negative, particularly for integers, thus showing that p(x) ≥ 0.

For n = 3, we have p(3) = x^3 + 4x + 3.

Factoring p(n): $p(n) = (x + 1)(x^2 + 3).$ Therefore, the factorized form is $(x + 1)(x^2 + 3).$

To find the inner and outer radii, we solve: $(x - a^2)^2 = b^2 - h^2$

Taking the square root: $x - a^2 = \pm \sqrt{b^2 - h^2}$ Hence: $x_1 = a^2 - \sqrt{b^2 - h^2}$
and
$x_2 = a^2 + \sqrt{b^2 - h^2}$.

The area of the annulus is given by: $\text{Area} = \pi (x_2^2 - x_1^2) = \pi \left((a^2 + \sqrt{b^2 - h^2})^2 - (a^2 - \sqrt{b^2 - h^2})^2\right).$

Simplifying, we find: $\text{Area} = 2\pi a^2\sqrt{b^2 - h^2}.$ This represents the area of the cross-section.

To find the volume of the torus: $V = \text{Area} \times 2\pi a.$

Substituting the area found previously: $V = 2\pi a^2\sqrt{b^2 - h^2} \times 2\pi a = 4\pi^2 a^3 \sqrt{b^2 - h^2}.$

Thus, the volume of the torus is given by: $V = 4\pi^2 a^3 \sqrt{b^2 - h^2}.$