An amusement park is planning to build a zip-line above a hill on its property.
The hill is modelled by `y = (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.
- Find `(dy)/(dx)`. (1 mark)
- 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 <= x <= a`, where `10 <= a <= 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 <= x <= 30`, as shown in the graph below.
- State the rule, in terms of `x`, for the height of the cable above the horizontal axis for `x in [a, 30]`. (1 mark)
- 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`.
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- State the gradient of the cable at `A`, in terms of `a`. (1 mark)
- Find the coordinates of `A`, with each value correct to two decimal places. (3 marks)
- Find the value of the gradient at `A`, correct to one decimal place. (1 mark)