How much time is there between the first low tide and the next high tide?

Find the difference in height between low tide and high tide.

Find the value of $p$;

Find the value of $q$;

Find the value of $r$.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

attempt to find the difference of $x$-values of A and B     (M1)

eg$\,\,\,\,\,$$6.25 - 12.5{\text{ }}$

6.25 (hours), (6 hours 15 minutes)     A1     N2

[2 marks]

attempt to find the difference of $y$-values of A and B     (M1)

eg$\,\,\,\,\,$$1.5 - 0.6$

$0.9{\text{ (m)}}$     A1     N2

[2 marks]

valid approach     (M1)

eg$\,\,\,\,\,$$\frac{{{\text{max}} - {\text{min}}}}{2},{\text{ }}0.9 \div 2$

$p = 0.45$     A1     N2

[2 marks]

METHOD 1

period $= 12.5$ (seen anywhere)     (A1)

valid approach (seen anywhere)     (M1)

eg$\,\,\,\,\,$${\text{period}} = \frac{{2\pi }}{b},{\text{ }}q = \frac{{2\pi }}{{{\text{period}}}},{\text{ }}\frac{{2\pi }}{{12.5}}$

0.502654

$q = \frac{{4\pi }}{{25}},{\text{ 0.503 }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)$     A1     N2

METHOD 2

attempt to use a coordinate to make an equation     (M1)

eg$\,\,\,\,\,$$p\cos (6.25q) + r = 0.6,{\text{ }}p\cos (12.5q) + r = 1.5$

correct substitution     (A1)

eg$\,\,\,\,\,$$0.45\cos (6.25q) + 1.05 = 0.6,{\text{ }}0.45\cos (12.5q) + 1.05 = 1.5$

0.502654

$q = \frac{{4\pi }}{{25}},{\text{ }}0.503{\text{ }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)$     A1     N2

[3 marks]

valid method to find $r$     (M1)

eg$\,\,\,\,\,$$\frac{{{\text{max}} + {\text{min}}}}{2},{\text{ }}0.6 + 0.45$

$r = 1.05$     A1     N2

[2 marks]

METHOD 1

attempt to find start or end $t$-values for 12 December     (M1)

eg$\,\,\,\,\,$$3 + 24,{\text{ }}t = 27,{\text{ }}t = 51$

finds $t$-value for second max     (A1)

$t = 50$

23:00 (or 11 pm)     A1     N3

METHOD 2 

valid approach to list either the times of high tides after 21:00 or the $t$-values of high tides after 21:00, showing at least two times     (M1) 

eg$\,\,\,\,\,$${\text{21:00}} + 12.5,{\text{ 21:00}} + 25,{\text{ }}12.5 + 12.5,{\text{ }}25 + 12.5$

correct time of first high tide on 12 December     (A1)

eg$\,\,\,\,\,$10:30 (or 10:30 am) 

time of second high tide = 23:00     A1     N3

METHOD 3

attempt to set their $h$ equal to 1.5     (M1)

eg$\,\,\,\,\,$$h(t) = 1.5,{\text{ }}0.45\cos \left( {\frac{{4\pi }}{{25}}t} \right) + 1.05 = 1.5$

correct working to find second max     (A1)

eg$\,\,\,\,\,$$0.503t = 8\pi ,{\text{ }}t = 50$

23:00 (or 11 pm)     A1     N3

[3 marks]