document.write( "Question 1182274: At this one coastal city, the tides vary greatly during the day. During May, the predicted height of the tides are 2.5 m at 4 AM, 17 m at 10 AM, 2.5 m at 4 PM and 17 m at 10 PM. Assume we start the graph at midnight. What is the height of the tide at 11 AM? Round to the nearest tenth. \n" ); document.write( "

Algebra.Com's Answer #812504 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "The minimum value is 2.5 at 4AM and the maximum value is 17 at 10AM.

\n" ); document.write( "So the period is 12 hours (because half the period -- from minimum value to maximum value -- is 6 hours) and the amplitude is (17-2.5)/2 = 7.25; the midline is then 2.5+7.25 = 9.75.

\n" ); document.write( "The problem says to assume the graph starts at midnight, but it never says anything about finding a function that fits the given data. This makes finding the answer to the given question much easier than it would otherwise have been.

\n" ); document.write( "The maximum value is at 10AM
\n" ); document.write( "The basic cosine function has its maximum value at 0, so we can consider our function a basic cosine function with t=0 at 10AM
\n" ); document.write( "11AM is 1 hour after the maximum
\n" ); document.write( "The period is 12 hours, so 11AM is 1/12 of a period after the maximum value, corresponding to an angle of pi/6 radians, or 30 degrees.
\n" ); document.write( "cos(30) = sqrt(3)/2

\n" ); document.write( "So the height of the tide at 11AM is (sqrt(3)/2) times the amplitude above the midline:

\n" ); document.write( " $\"9.75%2B7.25%28sqrt%283%29%2F2%29\"$

\n" ); document.write( "which to the nearest tenth is 16.0.

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