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The average depth of water at the end of a dock is 7 feet.
This varies 3 feet in both directions with the tide.
Suppose there is a high tide at 4a.m. if the tide goes from low to high every 6hrs.
(A) Write a cosine function d(t) describing the depth of the water as a function of time
with t = 4 corresponding to 4a.m.
(B) At what two times within one cycle is the tide at a depth of 8.5 feet ?
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They talk about a harmonic function of the height with the average value of 7 feet;
the maximum value of 7+3 = 10 ft at 4 am; and the period of 2*6 = 12 hours (= one cycle).
+------------------------------------------------------------+
| They ask you to write this function as a function of t, |
| placing t= 4 at t= 4 am. |
+------------------------------------------------------------+
From the description, it is clear that the function is cosine with the argument (t-4)
with no shift. So we write
h(t) = ft.
Part (A) is solved.
To solve part (B), write this equation as the problem requires
h(7) = 8.5 ft, or
= 8.5.
Simplify
= 8.5 - 7
= 1.5
=
=
= or .
Reduce the factor in both sides to get
= or .
Multiply by 12 both sides
2(t-4) = 4 or 20
2t - 8 = 4 or 20
2t = 12 or 28
t = 6 or 14 (military time)
ANSWER. Time then the depth is 8.5 ft is 6 am and/or 2 pm
(two time moments within each 12-hours cycle).
The solution is complete - both questions are answered.