MHF 4U1
APPLICATIONS OF SINUSOIDAL FUNCTIONS II
1 Tidal forces are greatest when the Earth, the sun, and the moon are in line.
When this occurs at the Annapolis Tidal Generating Station, the water has a
maximum depth of9.6 metres at 4:30 am and a minimum depth of 0.4 metres 6.2 hours later.
.
(a)
(b)
(c)
(d)
Quick sketch the application for depth of water d vs time K. Label all main points on both axes.
Write an equation for this sinusoidal function.
Calculate the depth ofthe water at 9:30 am and at 6:45 pm. Answer to 2 decimal places.
During this day when will the depth ofthe water be 3.5 m?
2. At St. John’s, Newfoundland the time ofthe sunrise on the n th day ofthe year is given by the formula:
t
=
1.89sin[--(n 80)] + 6.41, where I is the time in hours during the day.
_
(a) Quick sketch the application. Label all main points on both axes.
(b) Determine the time the sun rises on October 20 (day 293).
(c) On what day numbers ofthe year will the sun rise at 6: 15 am?
3. The depth ofwater, d(t) metres, in a seaport can be approximated by the sine function:
d(t) = 2.5cos[—L(t 4.55)] + 13.4
_
where
t
is the time in hours since midnight.
(a) Quick sketch the application. Label all main points on both axes.
(b) What is the depth ofthe water at 10 am? Answer to 2 decimal places.
(c) A cruise ship needs a depth ofat least 12 m ofwater to dock safely.
What is the longest stretch oftime in which the ship can dock safely? Answer to 1 decimal place.
4. On the n th day ofthe year, the number ofhours ofdaylight h(t) for Christchurch, New Zealand
is given by the formula:
hQ) = _3.98sinE28l+ 12.16
L 365 J
(a) Quick sketch the application. Label all main points on both axes.
th
(b) How many hours ofdaylight will there be on the 100
day? Answer to 2 decimal places.
(c) On what day numbers ofthe year will there be 10 hours of daylight?
5. A Ferris wheel has a radius of 25 metres, and its centre is 26 metres above the ground.
It rotates once every 50 seconds. Suppose a person boards the ferris wheel at t = 0 seconds.
(a) Quick sketch the application for the person’s height above ground h vs time t.
Label all main points on both axes.
(b) Write an equation for this sinusoidal function.
(c) How high will you be above the ground after 20 seconds? Answer to 2 decimal places.
(d) In the first 2 minutes how long are you 30 m above the ground? Final answer to 1 decimal place.
6. At a certain point on the beach, a post sticks out ofthe sand, its top being 76 centimeters above the beach.
The depth ofthe water at the post varies sinusoidally with time due to the motion ofthe tides.
The depth d, in centimeters is:
d = 6Ocos 2,r—--------- + 40 where t is the time in hours since midnight.
L
12
J
(a) Quick sketch the application for d vs I. Label all main points on both axes.
(b) What is the earliest time ofday at which the water level is just at the top ofthe post?
(c) At the time you calculated in part (b), is the post just going under water or
just emerging from the water?
(d) Between what times during this day will the entire post be out ofthe water?
76 cm
7. Tarzan is swinging back and forth on his grapevine.
As he swings, he goes back and forth across the river bank, going alternately over land and water.
Jane decides to mathematically model his motion and starts her stopwatch.
Let I be the number of seconds the stopwatch reads and let d be the number of meters Tarzan is from
the river bank. Assume that d varies sinusoidally with K, and that d is positive when Tarzan is over
water and negative when he is over land.
Jane finds that when I = 2, Tarzan is at one end ofhis swing and d = —23.
She finds that when I = 5, he reaches the other end ofhis swing and d 17.
—
(a) Quick sketch the application for Tarzan’s distance from the river bank, d vs time I.
Label all main points on both axes.
0’) Write an equation for this sinusoidal function.
(c) Where is Tarzan after: (i) 2.8 seconds? (ii) 6.3 seconds? Answers to 2 decimal places.
(d) Where was Tarzan when Jane started the stopwatch?
(e) At what times during the first 15 seconds oftiming will Tarzan be exactly lOm over land?
(answers correct to 2 decimal places)
Answers
1. (b) d =4.6cos[j(t_4.5)]+5
(c) 1.22 metres
7.72 metres
(d) 12:44am,8:l6am, 1:08pm, 8:40am
2. (b) 5:28 am
(c) day 75 and day 267
3. (b) 1 1 .04 metres
(c) 8.4 hours
4. (b) 10.82 hours
(c) day 1 13 and day 229
r2 1
5. (b) h=—25cosI—tj+26
[50 J
(c) 46.23 metres
(d) for 5 1 1 seconds
6. (b) 12:14am
(c) going under
(d) between 6:23 am and 9:37 am
.
AND
between 6:23 pm and 9:37 pm
,
2
r
7. (b) d20sinI_(t_3.5)]_3
[6
(c) (i) 16.38 metres from the bank, over land
(ii) I 1 6 metres from the bank, over water
.
(d) 7.00 metres from the bank, over water
(e) 0.84 seconds, 3.16 seconds, 6.84 seconds, 9.16 seconds, & 12.84 seconds
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