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Problem 7 (16 pts) The depth of a swimming salmon
below the water surface can be modeled by a
sinusoidal function of time. The salmon’s depth varies
between a minimum of 1 foot and a maximum of 7
feet below the surface of the water. It takes the
salmon 1.8 minutes to move from its minimum depth
to its successive maximum depth, and it first reaches
the minimum depth at
minutes.
a) Find the sinusoidal function
(
)
which models the depth of the salmon
after minutes.
b) Compute all the times during the first 5 minutes when the depth of the fish is exactly 3 feet.
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3. The temperature at your house in the desert is a sinusoidal function of time with a 24
hour period. The maximum daily temperature is 45 degrees Celsius and occurs at 5:00
PM. The minimum daily temperature is 11 degrees Celsius.
(a) Let t be hours after midnight last night. Find the function f (t) that gives the temperature at time t.
(b) For how much of each day is the temperature above 35 degrees Celsius?
(c) Starting from midnight last night, how long will it be until the temperature has been
above 35 degrees Celsius for 22 hours?
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5. The diameter of a certain cloud in the sky above Seattle is a sinusoidal function of time.
At 7 AM this morning, the diameter of the cloud was at its minimum, 20 meters.
The cloud then expanded, and reached its maximum diameter of 26 meters at 11:30 AM
this morning.
From 3 AM this morning to 3 PM this afternoon, for how much time was the cloud’s
diameter less than 21.5 meters?
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Assume that the depth of the shore at Neah Bay is
given by the sinusoidal function
π
d(t) = 12 sin( (t − 3)) + 15
6
where t is measured in hours (after Jan 1st 2014) and
d is measured in feet.
1. What is the maximum width of the beach and when
it is reached ?
2. What is the minimum width of the beach and when
is it reached ?
3. When on Jan 1st 2014 (0 ≤ t ≤ 23) is the width
of the beach 10 feet ?
4. When on Jan 2nd 2014 is the width of the beach
10 feet ?
5. For how long in the morning of Jan 1st ((6 ≤ t ≤
12)is the width of the beach greater than 10 feet?
1
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