After 8 minutes, the seat is 117 m above the ground. Find k.

Find the value of a.

Find when the seat is 30 m above the ground for the third time.

valid approach to find k      (M1)

eg   8 minutes is half a turn, k + diameter, k + 111 = 117

k = 6      A1 N2

[2 marks]

METHOD 1

valid approach      (M1)
eg  $\frac{{{\text{max}}\,\, - \,\,{\text{min}}}}{2}$ a = radius

$\left| a \right| = \frac{{117 - 6}}{2},\,\,55.5$     (A1)

a = −55.5      A1 N2

METHOD 2

attempt to substitute valid point into equation for f      (M1)
eg  h(0) = 6, h(8) = 117

correct equation      (A1)
eg   $6 = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8} \times 0} \right),\,\,117 = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8} \times 8} \right),\,\,6 = 61.5 + a$

a = −55.5      A1 N2

[3 marks]

valid approach      (M1)
eg   sketch of h and $y = 30,\,\,h = 30,\,\,61.5 - 55.5\,{\text{cos}}\left( {\frac{\pi }{8}t} \right) = 30,\,\,t = 2.46307,\,\,t = 13.5369$

18.4630

t = 18.5 (minutes)      A1 N3

[3 marks]