Researchers find a creature from an alien planet. Its body temperature varies sinusoidally with time. It reaches a high of $120^o F$ in $35$ minutes. It reaches a low of $104^{o}F$ in $55$ minutes. a) Sketch a graph b) Write an equation expressing temperature in terms of minutes since they started timing c) What was its temperature when they first started timing? d) Find the first three times after they started timing at which the temperature was at $114$
Hi, so I'm stuck on the first letter. I've got the equation $-47\sin(\frac{\pi}{20})+167$ thus far. You see, for the period, I reasoned that the temperature must range in $40$ minutes, right? Because if $120$ is the high and $104$ is the low, $35$ and $55$ respectively, then if you place $120$ at the top of a circle and 104 at the bottom (sinusoidal functions are basically circles) and then you can tell a $20$ minute difference. A full circle would be $40$ minutes. I understand that -- but this also means there has to be a horizontal shift, I think. How do I find the horizontal shift without using a calculator, assuming I know the period? Even if all of the work I did was wrong thus far, I would still like to know how to find the horizontal shift without using a calculator. Thanks!
Your reasoning that the period is $40$ minutes is correct. Your equation is not correct. Note that the sine ranges from $-1$ to $+1$, so the multiplier of the sine function is half the peak-to-peak range. The sine averages to $0$, so the constant should be halfway between the maximum and minimum. You also need a time offset in the argument of the sine function. The sine is maximum at $\frac \pi 2$ and your first maximum is at $35$ minutes, so you need the argument to be $\frac \pi 2$ at $35$ minutes. Just add in a constant to make that so.