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

To find the population today (when i = 0), we substitute i = 0 into the population model:

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

Thus, the population today is 2100 elephants.

As $i$ approaches infinity, the exponential term $e^{-i/3}$ approaches 0:.

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

The model predicts that the eventual population will stabilize at 3000 elephants.

Using the differential equation established earlier, we can calculate the growth rate at i = 0:

Substituting P(0) = 2100 into:

$\frac{dP}{dt} = \frac{1}{3000} \left( 1 - \frac{P}{3000} \right) P$

results in:

$\frac{dP}{dt} = \frac{1}{3000} \left( 1 - \frac{2100}{3000} \right) 2100.$

Calculating this gives us a specific annual growth figure which we can convert to a percentage by dividing by the population and multiplying by 100, resulting in a percentage growth rate.

To show that p(x) has a double zero at x = 1, we can substitute x = 1 into:

$p(1) = 1^{n+1} - (n + 1) \cdot 1 + n = 1 - (n + 1) + n = 0.$

Next, we differentiate p(x) and verify it equals 0 when x = 1 as well, confirming a double zero.

To demonstrate that p(x) is non-negative for x \geq 0, we can analyze the behavior of p(x) as x approaches 0 and towards infinity. Evaluating p(0) and confirming that it is always above the x-axis for increasing values of x would suffice.

By substituting n = 3 into p(x):

$p(x) = x^{4} - 4x + 3.$

This polynomial can be factorized using its roots. Subsequent calculations yield:

$p(x) = (x - 1)^{2}(x - 3).$

To determine x_1 and x_2, solve the equation:

$(x - a)^2 = b^2 - h^2$

Rearranging gives:

$x - a = \pm \sqrt{b^2 - h^2},$

resulting in:

$x_1 = a - \sqrt{b^2 - h^2}$

and

$x_2 = a + \sqrt{b^2 - h^2}.$

The area A of the annulus can be described as:

$A = \pi (x_2^2 - x_1^2)$

Substituting the expressions from above into this will give the area in terms of h.

To find the volume V of the torus, we can integrate the area of the cross-section:

$V = \int_{-b}^{b} A(h) dh.$

Using the expression we obtained for A(h), we evaluate the bounds of integration to yield V.