A bungee jumper of height 2 m falls from a bridge which is 125 m above the surface of the water, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 7 - 2009 - Paper 1

To find the length L of the cord such that the jumper's velocity is 30 m/s when $x = L$, we set the velocity function obtained from the previous integration to 30 m/s:

1. Solve for r:
   • 30 = Substitute the respective known values into the velocity equation, which includes previously calculated values of $g$ and v.
2. Apply numerical methods if necessary to solve for L with appropriate values, ensuring answers are given to two significant figures.

In this stage, we examine the equation:

$x = e^{\frac{t}{10}} (29 \sin t - 10 \cos t) + 92$

1. Plug in $t$ values that represent when the jumper's feet pass through $x = L$. Calculate the position.

2. Compare the calculated position against the height of the bridge (125 m) to determine if $x$ is greater than 125 m.

3. If the result shows $x$ exceeds 125 m, the jumper's head does stay out of the water; otherwise, it does not.

To show that:

1. Start with the expression:
   • Let: $z = \cos \theta + i \sin \theta$
   • Then the conjugate is $z^* = \cos \theta - i \sin \theta$
2. Calculate:
   • $z^2 = (\cos \theta + i \sin \theta)^2$ and $z^{*2} = (\cos \theta - i \sin \theta)^2$
3. Combine to evaluate:
   • Add the two results to simplify to the required form of $2 \cos n\theta$.

1. Use the results of part i to derive this expression by expanding using trigonometric identities. 2. Collate terms and utilize $m$ in the equation for systematic summation. 3. Observe and confirm the pattern emerges as required.

1. Utilizing the integration from previous steps, setup the definite integral for evaluation. 2. Implement symmetry arguments if possible. 3. Solve the integral through known integration methods and techniques, yielding the required result.