VCE Methods Differential Calculus Application Task 11

Question 1 [2017 Exam 2 Section B Q2]

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point \(P\). The height of \(P\) above the ground, \(h\), is modelled by \(h(t) = 65 - 55\cos\left(\frac{\pi t}{15}\right)\), where \(t\) is the time in minutes after Sammy enters the capsule and \(h\) is measured in metres.
Sammy exits the capsule after one complete rotation of the Ferris wheel.

a. State the minimum and maximum heights of \(P\) above the ground. 1 mark

b. For how much time is Sammy in the capsule? 1 mark

c. Find the rate of change of \(h\) with respect to \(t\) and, hence, state the value of \(t\) at which the rate of change of \(h\) is at its maximum. 2 marks

d. As the Ferris wheel rotates, a stationary boat at \(B\), on a nearby river, first becomes visible at point \(P_1\). \(B\) is 500 m horizontally from the vertical axis through the centre \(C\) of the Ferris wheel and angle \(CBO = \theta\), as shown below.

Find \(\theta\) in degrees, correct to two decimal places. 1 mark

Part of the path of \(P\) is given by \(y = \sqrt{3025 - x^2} + 65, x \in [-55, 55]\), where \(x\) and \(y\) are in metres.

e. Find \(\frac{dy}{dx}\). 1 mark

As the Ferris wheel continues to rotate, the boat at \(B\) is no longer visible from the point \(P_2(u, v)\) onwards. The line through \(B\) and \(P_2\) is tangent to the path of \(P\), where angle \(OBP_2 = \alpha\).

f. Find the gradient of the line segment \(P_2B\) in terms of \(u\) and, hence, find the coordinates of \(P_2\), correct to two decimal places. 3 marks

g. Find \(\alpha\) in degrees, correct to two decimal places. 1 mark

h. Hence or otherwise, find the length of time, to the nearest minute, during which the boat at \(B\) is visible. 2 marks