Q1: A particle moves in a straight line such that it passes through a fixed point O at time
t = 0, where t represents time measured in seconds after passing O. For 0 ≤ t ≤ 10
Its velocity, v meters per second, is given by
v = 2sin(0.5t) + 0.3t – 2
The graph of v is shown in the following diagram.
(a) Find the smallest value of t when the particle changes direction.
The displacement of the particle is measured in metres from O.
(b) Find the range of values of t for which the displacement of the particle is increasing.
(c) Find the displacement of the particle relative to O when t = 10
Q2: Sule Skerry and Rockall are small islands in the Atlantic Ocean, in the same time
zone.
On a given day, the height of water in metres at Sule Skerry is modelled by the function
H(t) = 1.63sin(0.513(t – 8.20)) + 2.13, where t is the number of hours after midnight.
The following graph shows the height of the water for 15 hours, starting at midnight. At
low tide the height of the water is 0.50m. At high tide the height of the water is 3.76m.
a) The length of time between the first low tide and the first high tide is 6 hours
and m minutes. Find the value of m to the nearest integer.
(b) Between two consecutive high tides, determine the length of time, in hours, for which
the height of the water is less than 1 meter.
(c) Find the rate of change of the height of the water when t = 13, giving your answer in
metres per hour.
On the same day, the height of water at the second island, Rockall, is modelled by the
function h(t) = asin(b(t – c)) + d, where t is the number of hours after midnight.
and a, b, c, d > 0.
The first low tide occurs at 02:41 when the height of the water is 0.40 m.
The first high tide occurs at 09:02, when the height of the water is 2.74 m.
(d) Find the values of a, b, c, and d.
When t = T, the height of the water at Sule Skerry is the same as the height of the water
at Rockall for the first time.
(e) Find the value of T.
Q3: The following graph shows the depth of water, d meters, at a point P, during one
day. The time t is given in hours, from midnight to noon.
a. Use the graph to write down an estimate of the value t when:
i. the depth of water is minimum,
ii. the depth of water is maximum,
iii. the depth of water is increasing most rapidly.
b. The depth of water can be modeled by the function
d = A cos(B(t – C)) + D.
i. Show that A = 8.
ii. Write down the value of D.
iii. Find the value of B.
c. A sailor knows that he cannot sail past point P when the depth of water is less than
12 m. Calculate the values of t between which he cannot sail past P.
Q4: a. Sketch the graph of f(x) = 3cos(x) – 1 on the interval 0° ≤ x ≤ 360°.
b. Sketch the graph of g(x) = 5cos(x) over the same domain.
c. Hence, find the value of x where the two graphs meet.
Q5: A scientist is measuring the intensity of a sound wave emitted by a rotating
mechanical siren. The intensity I, measured in decibels (dB), is modeled by the formula:
I(t) = 10 log₁ ₀ (P(t))
where P(t) = 5 + 3sin(πt / 6) represents the power of the sound in watts at time t, in
hours after midnight.
a) Sketch the graph of I(t) for 0 ≤ t ≤ 12.
b) Find the maximum and minimum sound intensity (in dB) during this period.
c) At what time(s) is the sound intensity increasing most rapidly?
d) A hospital nearby must act if sound levels exceed 10 dB. Determine the
interval(s) of time during which the sound level exceeds this threshold.
Q6: A sound engineer is analyzing the pressure variation of a sound wave produced by
a tuning fork. The sound wave can be modeled by the function:
p(t) = 0.02 * sin(880πt)
where p(t) is the pressure variation in Pascals, and t is the time in seconds.
a) What is the amplitude and period of the wave? Interpret their meaning in this
context.
b) Sketch the graph of p(t) for 0 ≤ t ≤ 0.01.
c) How many complete sound wave cycles occur in the interval 0 ≤ t ≤ 0.01
seconds?
d) If the ear can detect pressure variations above 0.015 Pascals, find the time
intervals within 0 ≤ t ≤ 0.01 when the sound is audible.
Q7: A surveillance drone is flying at a constant altitude and records its position at two
different times.
At time t seconds, the angle of elevation θ (in degrees) from a fixed point A on the
ground to the drone is given by:
θ(t) = 45 + 15 sin(πt / 20)
The distance d (in meters) from the drone to the fixed point A is recorded every 5
seconds and is modeled by:
d(t) = 1000 × 10^(−0.02t)
a) Determine the maximum and minimum angle of elevation during the flight.
b) Calculate the drone’s distance from point A at t = 0 and t = 60 seconds.
c) Using both equations, write an expression for the vertical height h(t) of the drone
above point A in terms of θ(t) and d(t).
d) Use your expression to find the vertical height of the drone at t = 0 and t = 60
seconds.
e) Explain how the trigonometric and logarithmic components together model a realistic
drone movement scenario.