VCE Methods Differential Calculus Application Task 8

Question 1 [2019 Exam 2 Section B Q2]

An amusement park is planning to build a zip-line above a hill on its property.
The hill is modelled by \(y = \frac{3x(x-30)^2}{2000}\), \(x \in [0, 30]\), where \(x\) is the horizontal distance, in metres, from an origin and \(y\) is the height, in metres, above this origin, as shown in the graph below.

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

b. State the set of values for which the gradient of the hill is strictly decreasing. 1 mark

The cable for the zip-line is connected to a pole at the origin at a height of 10 m and is straight for \(0 \le x \le a\), where \(10 \le a \le 20\). The straight section joins the curved section at \(A(a, b)\). The cable is then exactly 3 m vertically above the hill from \(a \le x \le 30\), as shown in the graph below.

c. State the rule, in terms of \(x\), for the height of the cable above the horizontal axis for \(x \in [a, 30]\). 1 mark

d. Find the values of \(x\) for which the gradient of the cable is equal to the average gradient of the hill for \(x \in [10, 30]\). 3 marks

The gradients of the straight and curved sections of the cable approach the same value at \(x=a\), so there is a continuous and smooth join at \(A\).

e.

i. State the gradient of the cable at \(A\), in terms of \(a\). 1 mark

ii. Find the coordinates of \(A\), with each value correct to two decimal places. 3 marks

iii. Find the value of the gradient at \(A\), correct to one decimal place. 1 mark