The wind speed at a weather monitoring station varies according to the function
`v(t) = 20 + 16sin ((pi t)/(14))`
where `v` is the speed of the wind, in kilometres per hour (km/h), and `t` is the time, in minutes, after 9 am.
- What is the amplitude and the period of `v(t)`? (2 marks)
- What are the maximum and minimum wind speeds at the weather monitoring station? (1 mark)
- Find `v(60)`, correct to four decimal places. (1 mark)
- Find the average value of `v(t)` for the first 60 minutes, correct to two decimal places. (2 marks)
A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function
`v_1(t) = 28 + 18 sin((pi(t - k))/(7))`
where `v_1` is the speed of the wind, in kilometres per hour, `t` is the time, in minutes, after 9 am and `k ∈ R^+`. The wind speed after 9 am is shown in the diagram below.
- Find the smallest value of `k`, correct to four decimal places, such that `v(t)` and `v_1(t)` are equal and are both increasing at 10 am. (2 marks)
- Another possible value of `k` was found to be 31.4358
Using this value of `k`, the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.
i. Find the value of `t` at which a signal is first sent, correct to two decimal places. (1 mark)
ii. Find the proportion of one cycle, to the nearest whole percent, for which `v_1 > 38`. (2 marks)
- Let `f(x) = 20 + 16 sin ((pi x)/(14))` and `g(x) = 28 + 18 sin ((pi(x - k))/(7))`.
The transformation `T([(x),(y)]) = [(a \ \ \ \ 0),(0 \ \ \ \ b)][(x),(y)] + [(c),(d)]` maps the graph of `f` onto the graph of `g`.State the values of `a`, `b`, `c` and `d`, in terms of `k` where appropriate. (3 marks)