SOLUTION: The temperature inside a greenhouse varies dependent on the time of day. The maximum temperature of 32°C was at 1:00pm and the minimum temperature was 25°C at 1:00am. Assuming th

Algebra.Com
Question 725618: The temperature inside a greenhouse varies dependent on the time of day. The
maximum temperature of 32°C was at 1:00pm and the minimum temperature was
25°C at 1:00am. Assuming the temperature in the greenhouse can be modelled
using a sine function, find the equation that would model the greenhouse
temperature from midday.
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
The equation we will write will be of the form:
y = C + A*sin(B(x-D))
where
| A | is the amplitude
B is a number related to the period by the equation B = 2pi/period
C is the vertical shift
D is the phase (aka horizontal) shift.

All we have to do is take the information from the problem and use it to figure out correct values for A, B, C and D.

With a maximum temperature of 32 and a minimum of 25, the halfway point would be 28.5. This is the vertical shift. This is C.

From the halfway point to the maximum (or minimum) is 3.5. This is the amplitude. This is A.

The cycle repeats every 24 hours so the period is 24 hours. This makes


The normal sin function starts at the halfway point and then moves from there up to the maximum. We have to find where this should be in order for the graph to start at noon (as the problem instructs). So let's piece this together. At the y-axis, the time is noon (as the problem says it should be). One hour later, 1 pm, we have the maximum temperature. Twelve hours later, 1 am, we have the minimum temperature. 1/4 of a cycle later we will be back to the halfway point. (It is always 1/4 of a cycle from one key point to the next: from a maximum to the next halfway point, from the halfway point to the minimum, etc.) Since the period is 24 hours, 1/4 of a cycle is 6 hours. So 6 hours after the minimum we will be at the halfway point on our way to the next maximum. This the the point we've been looking for. It took us 1 hour to the maximum, 12 hours to the minimum and finally 6 hours to this point. All together it took 19 hours. This is our phase shift. This is D.

We're finally ready to write our equation:


P.S. The problem says "... find the equation ...". This is not possible. There is not just one equation that will work. There are literally an infinite number of equations that will work. In addition to the one we found we could use:FWIW, here's what the graph looks like:

Note: The x values represent "hours since noon".
RELATED QUESTIONS

The temperature inside Daphne's refrigerator was 10 degrees at 1:00PM. If the temperature (answered by Alan3354)
One day the temperature at 8:00 am was 41F and by 1:00Pm the temperature was 61F. What... (answered by Zoop)
The temperature at, T (degrees Celsius), inside a building on given day is given by the... (answered by solver91311)
The lowest temperature on a winter morning was -8F. Later that same day the temperature... (answered by rapaljer)
One day, the temperature at 7:00 a.m. was 41F, and by 1:00 p.m. the temperature was 59F.... (answered by Nate)
Does the temperature outside depend on the time of day, or does the time of day depend on (answered by Boreal)
4. One day, the temperature at 5:00 A.M. was 45°F, and by 1:00 P.M. the temperature was... (answered by Nate)
The temperature of a patient during an illness is given by the function​ below,... (answered by Boreal)
One day, the temperature at 2:00 A.M. was 43F, and by 3:00 the temperature was 56 F.... (answered by Nate)