Find Function of Time(t) — Trigonometry

Question

The water at a boat dock is 7 feet deep at low tide, and 11 feet deep at high tide.

One day, low tide occurred at 4 am and high tide at 10 am.

Find a function for the height of the tide as a function of time t .

Think about the characteristics: amplitude, period, midline, and try to sketch a figure reflecting how the tide height moves up and down.

Answer 1

Height H at 4am = 7 ft T = time in hours

Height at 10 am = 11 ft

Find the slope ..... delta H / delta T = slope = ( 11 - 7 ) / ( 10 - 4 ) = 4 / 6 = 2/3

We can see there are 2 points on the Graph of H vs T points ( T, H )

Since T is the independent variable [ like x for y vs x ] , so it comes first.

The points are ....

( 4 , 7 ) and ( 10 , 11 )

Now use the point slope formula

H - H1 = m ( T - T1 )

H - 7 = 2/3 ( T - 4 )

H = 2/3 T - 8/3 + 7

H = 2/3 T + 13/3 for 4 < T < 10

Now check the points with the equation

If T = 4 H = 2/3 (4) + 13/3 = 21/3 = 7 check ok

If T = 10 H = 2/3 (10) + 13/3 = 33/3 = 11 checks ok