1. A small windmill has its center 6 m above the ground and blades 2 m
in length. In a steady wind, a point P at the tip of one blade makes a
complete rotation in 12 s.
a) If the rotation begins at the highest possible point, draw a sketch
showing 1 cycle.
b) Determine a function that gives the height of point P above the
ground at a time t.
c) What is the height of point P at 5 s and 40 s? Check your answer for 5
s by looking at the graph.
d) At what time during the first cycle is P exactly 7 m above the ground?
Confirm your answer by looking at the graph.
2. The function S = –3 cos (30t) + 20 represents the surface temperature,
S, of a pond, in degrees Celsius, and t represents the numbers of hours
since sunrise at 06:00. At what times is the surface temperature of the
water 20oC between 6 am and 6 pm?
3. A water wheel has a radius of 3 m and has 0.5 m submerged below
the water. It rotates every 40 s.
a) Draw the graph of the height of a point P on the wheel above (or
below) the water for 2 complete cycles, starting at the lowest point.
b) Find the equation of this graph.
c) Find the height of point P at t = 32 s.
4. In biology the predator-prey model is used to predict the number of
prey animals, R, at a particular time, t, in years. This number is cyclic,
fluctuating with the number of predators. The number of rabbits, R, in a
certain region is given by the equation R = 2000 sin (120t) + 5000.
a) Sketch the graph of this relationship
b) Determine how many rabbits there will be in 4 years.
5. At a certain point on the beach, a post sticks out of the sand. The top
of the post is 75 cm above the beach. The depth of the water at the post
varies with time due to the tide. If h is the height of the water t hours
after midnight, the equation giving the water height is
h = 40 cos [30(t – 3] + 50.
a) Sketch the graph of this function
b) What is the depth of the water at 11AM?
c) At what time of the day is the tide at its lowest point?
d) During the first 12 hours, when is the post under water? Confirm your answer
using the graph.
Answers
1. b) h = 2 cos (30t) + 6
c) 4.3 m, 5 m
d) 2s, 10s
2. 9 AM and 3 PM
3. b) h = – 3 cos(9t) + 2.5
c) 1.6 m
4. 6732
5. b) 30 cm c) 9 am
d) Between 1.3 and 4.7 hours (between 1:18 AM and 4:.12 AM)