Question 3 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 3 - 2002 - Paper 1

First, we calculate the total seating arrangements without restrictions, which is 720 as shown earlier.

Next, we need to find the arrangements where Kevin and Jill sit together. We can treat Kevin and Jill as a single unit or block. This means we now have 6 units to arrange (Kevin-Jill block + 5 other people).

The number of arrangements for these 6 units in a circle is: $5! = 120.$

Within the Kevin-Jill block, there are 2 arrangements (Kevin can sit to the left of Jill or to the right).

Therefore, the number of arrangements where Kevin and Jill sit together is: $5! \times 2 = 120 \times 2 = 240.$

Finally, to find the number of arrangements where they do not sit together, we subtract the cases where they sit together from the total arrangements: $720 - 240 = 480.$

So, there are 480 seating arrangements possible.

To show that $f(x)$ has a root between $x = 3.7$ and $x = 3.8$, we evaluate the function at these points:

• For $x = 3.7$: $f(3.7) = e^{-3.7} - 3(3.7)^2$

• For $x = 3.8$: $f(3.8) = e^{-3.8} - 3(3.8)^2$

Calculating the values allows us to check if $f(3.7)$ and $f(3.8)$ have opposite signs, indicating a root exists by the Intermediate Value Theorem.

Newton's method updates a guess of a root using the formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

First, we need to compute $f'(x) = -e^{-x} - 6x$.

We compute:

• $f(3.8)$ and $f'(3.8)$. Then, we substitute $x=3.8$ into the Newton's formula for one iteration to get a better approximation.

To verify that the proposed solution is valid, we differentiate $T$ with respect to $t$:

$\frac{dT}{dt} = -kAe^{-kt}.$

Substituting $T$ back into the original differential equation to check if both sides equal verifies the solution.

Using the initial condition, when $t = 0$, $T = 80$: $80 = 22 + A$ Thus, $A = 58$. Next, we find $k$ using the condition that $T$ is 60 after 10 minutes: $60 = 22 + 58e^{-10k}.$

We solve this equation for $k$.

We set up the equation: $30 = 22 + 58e^{-kt}.$

By rearranging and solving for $t$, we find the time required to reach 30°C. We'll round our answer to the nearest minute.