The high and low tides of a coastal city on May 3 2000 are s

The high and low tides of a coastal city on May 3, 2000 are shown in the table. Find the height of the water at 9:0.5 A. M. (Assuming that the water height can be modeled by a cosine function). If needed, round to three decimal places. The electrical current I, measured in amperes, flowing through a circuit at time t is modeled by the formula I = 110 sin 80 pi t What is the period? (justify your answer by showing calculations) What is the amplitude? (no need to show work on this one) Sketch one period. Be sure to correctly label the axes and all relevant points

Solution

Ans(14):

midpoint: (98 + 42) / 2 = 140 / 2 = 70

amplitude: |70 - 42| = 28

Hence basic formula in x for the height is:
28 cos(x) + 70

However, t starts at midnight, so given the value at high and low, we need to find the y intercept for t = 0, which will be the starting angle, and the cycle of the cosine wave 2 expressed as a function of t.

That will give us the exact wave

The time for a full 2 period in a cosine wave is high to low, to high, which is twice (low - high).
2 period: 2(8:27 - 2:00) = 2(6:27) = 12:54 or 12+54/60 hours = 12.9 hours

The angular movement in one hour is 2/12.9 = /6.45 = 0.487068628464

At -2 hours, it is -2/6.45 = -0.974137256927
So, at midnight, the angle is 0.974
and for each t, increases by 0.487
So the angle is 0.487t-0.974

The final equation: h(t)=27cos(0.487t-0.974)+70

now we have to find height at 9:05AM

means plug t=9+5/60=9.083

So plug t=9.083 in h(t)

h(t)=27cos(0.487*9.083-0.974)+70= 44.269

Hence answer is 44.269 cm