Question
Alex only swims in the sea if the water temperature is at least 15$^{\circ}$C. Alex goes into the sea close to home for the first time each year at the start of May when the water becomes warm enough.
Alex models the water temperature at midday with the function $f(x) = asinbx + c$ for $0 \leq x \leq 12$, where $x$ is the number of months after 1st May and where $a, b, c > 0$.
The graph of $y = f(x)$ is shown in the following diagram.
(a) Show that $b = \frac{\pi}{6}$.
(b) Write down the value of
(i) $a$;
(ii) $c$.
Alex is going on holiday and models the water temperature at midday in the sea at the holiday destination with the function $g(x) = 3.5\sin\frac{\pi}{6}x + 11$, where $0 \leq x \leq 12$ and $x$ is the number of months after 1st May.
(c) Using this new model $g(x)$
(i) find the midday water temperature on 1st October, five months after 1st May.
(ii) show that the midday water temperature is never warm enough for Alex to swim.
(d) Alex compares the two models and finds that $g(x) = 0.7f(x) + q$. Determine the value of $q$.
▶️Answer/Explanation
Solution:-
(a) $12b = 2\pi$ OR $(b = ) \frac{2\pi}{12}$ OR $12 = \frac{2\pi}{b}$
$b = \frac{\pi}{6}$
(b) (i) $a = 5$
(ii) $c = 15$
(c) (i) attempt to substitute $x=5$ into $g(x)$
$g(5) = 3.5 sin \frac{5\pi}{6} + 11$
$\sin \frac{5\pi}{6} = \frac{1}{2}$
$g(5) = 3.5 \times \frac{1}{2} + 11$
$g(5) = 12.75 = \frac{51}{4}$
(ii) METHOD 1 (finding maximum temperature)
considering the maximum value of $\sin \frac{\pi}{6}x (=1)$ OR $g'(x) = 0$ at maximum
OR maximum = vertical shift + amplitude (may be seen on a graph)
$g_{max} = 3.5 + 11$ OR $\frac{\pi}{6} \cdot 3.5 cos(\frac{\pi}{6}x) = 0$ OR $x = 3$
$g_{max} = 14.5$
$14.5 < 15$ (hence the midday water temperature is never warm enough for Alex to swim)
METHOD 2 (working with inequality)
$3.5sin\left(\frac{\pi}{6}x\right)+11 \geq 15$
$sin\left(\frac{\pi}{6}x\right) \geq \frac{8}{7}$
sine values can never be greater than 1 (hence the midday water temperature is never warm enough for Alex to swim)
(d) EITHER
attempt to find $0.7f(x)$ OR $0.7f(x) + q$
$0.7f(x) = 3.5sin\frac{\pi}{6}x + 10.5$
OR
$0.7f(x) + q = 3.5sin\frac{\pi}{6}x + 10.5 + q$
OR
$10.5 + q = 11$
OR
attempt to find $0.7f(x)$ for a particular value of $x$
eg maximum $20 \times 0.7 = 14$
THEN
$q = 0.5$