Above or below a level

When, and for how long

In 4 of 20 recent exams
Paper 1 or 2
2 to 5 marks

How it is set in the exam

mostly Paper 2 (calculator) (GDC), 2–5 marks: the last parts of a sinusoid question, once the model is known. Typical (4 of 20 exam versions: 1 on Paper 1 (no calculator), 3 on Paper 2 (calculator)). The Paper 1 (no calculator) version needs an exact trig equation; the Paper 2 (calculator) versions ask for a first time, a duration within one cycle, a total time over two cycles, or a proportion of time given as a probability.

Spot it in the exam

Exam phrases to spot (4)

“for how long is the height above 10 m?”
“during one revolution, for how many minutes …”
“find the first time at which the height is $6$ m”
“at a random time, find the probability that …”

In disguise (2)

"under water", "in the shade", "too hot to …": a level in words, where you decide which side of it is asked
a percentage of the time, or a probability: the time on the right side of the level, divided by the whole time

Variants

the first time (GDC); the duration in one period ( exact; GDC); the total over two or more cycles, including a cycle cut short by the end of the interval; the proportion of time as a probability.

Looks like it, but isn't (4)

the model equals another model, or a line that isn't horizontal: Solve graphically
a trig equation with no context, on an interval: Solve a trig equation
a quadratic model above a level: Quadratic inequalities
a turning object with no height formula (a radar beam or a sprinkler reaching a straight edge): the angle swept across a chord and the angular speed give the time: Chords, segments and regions

Worked example

A water wheel: how long each turn is under water

Paper 1 (no calculator)

Question

A water wheel of radius $4$ m turns at a constant rate, making one complete turn every $16$ seconds. Its centre is $2$ m above the surface of the water. The height of a point $P$ on the rim above the surface, $t$ seconds after $P$ is at its lowest point, is $h(t) = 2 - 4\cos\left(\frac{\pi t}{8}\right)$ metres. (A negative height is below the surface.)
(a) Find the values of $t$ , for $0 \le t \le 16$ , at which $P$ is at the surface of the water. [4]
(b) Hence find the length of time during each turn for which $P$ is under the water. [2]

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Steps

Sketch one period
of the model with the line $y = k$ , and mark which side is asked for (above or below).
Solve model $= k$ .
On , use zeros and intersections: write the equation and the interval, then every solution. On , isolate the sine or cosine, find the exact angles for the bracket's interval (Solve a trig equation), then the times. Solving with $=$ is accepted even when the question says "above", "more than" or "at least": the crossings are the same, and the sketch decides which side you keep.
Take consecutive crossings
that bound the part asked for. When the interval covers the whole cycle, a part that wraps round the end of the cycle is the period minus the other part; when the interval stops before the cycle ends, add the two end pieces instead (Step 4).
Several cycles, or less than one:
the same duration comes back every period, but check the last (or only) cycle: the interval may end part-way through it.
A proportion or a probability
is $\dfrac{\text{time on the side asked for}}{\text{whole time}}$ .
Answer in context,
with units; a first time as a clock time if asked.

Common mistakes

Only one crossing
the time above (or below) a level runs between two crossings, or from an end of the domain to a crossing (Example 3.1, Your turn 2 and 3); one time on its own is not an interval.
Counting one cycle
when the domain covers two or more, or assuming the last cycle is complete (Your turn 2 and 3).
The wrong side
"under the water" is $h < 0$ . Above and below swap the answer with the rest of the period.

More mistakes (5)

Doubling the first piece
when the interval starts inside the part asked for: the piece at the other end may be cut short.
Subtracting rounded crossings
use the stored values (keep full-accuracy values).
Decimal hours as a clock time
$10.596$ hours is 10:36, not 10:60 or 10:59.
The wrong whole time for a probability
divide by the whole interval ( $48$ hours), not by one period.
Degree mode
on : the model is in radians. Check the settings.

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