Sinusoidal Graph question

Hey everyone, I'm at a question that I'm a little stumped on the method of solving, and there is no solution guide in the textbook :(.

4. In a New Zealand city the time of sunrise can be found using the trigonometric function

T= 4.5sin(pi(d-4)/180)+4.9

where T is the number of hours past midnight d and is the number of days into the year. Find the time to the nearest minute when the

Sun will rise on the 7th day of the year. **How many days in the year does the sunrise after 7 am?**

The bolded part of the question is the part I am not sure how to solve.

Any help? :D

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Re: Sinusoidal Graph question

Days of the year when sunrise is after 7 a.m. are:

{32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156}

which are 125 days.

http://mathhelpforum.com/attachment....1&d=1348808188

This can't be NZ(Shake)

Re: Sinusoidal Graph question

You want to solve:

$\displaystyle 4.5\sin\left(\frac{\pi(d-4)}{180} \right)+4.9>7$

$\displaystyle 4.5\sin\left(\frac{\pi(d-4)}{180} \right)>2.1$

$\displaystyle \sin\left(\frac{\pi(d-4)}{180} \right)>\frac{7}{15}$

So we want:

$\displaystyle \sin^{-1}\left(\frac{7}{15} \right)<\frac{\pi(d-4)}{180}<\pi-\sin^{-1}\left(\frac{7}{15} \right)$

$\displaystyle \frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)<d-4<180-\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)$

$\displaystyle \frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)+4<d<184-\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)$

You should find, using your calculator that (approximately):

$\displaystyle 31.8<d<156.2$

Since *d* is an integer, we may write:

$\displaystyle 32\le d\le156$

Hence, the number of days *N* is:

$\displaystyle N=(156-32)+1=125$