I was given a graph and told to find an equation for the temperature T in terms of the time t in the form T(t) = a sin(kt-c)+d where a, k, c and d are constants.
The max was 32.9 and the min 21.1
I determined that
amplitude = 5.9
vertical = 27
period = pi/12
The last thing I have to find is phase shift please help!
I agree with a = 5.9
d = 32.9 - 5.9 = 27 , you had that
so forget the phase shift for the time being.
testing your equation:
t(t) = 5.9 sin (π/12t) + 27
32.9 = 5.9sin(π/12t) + 27
sin(12t/π) = (32.9-27)/5.9 = 1
12t/π = π/2
t = (π/2)(12/π) = 6
21.1 = 5.9sin(12t/π) + 27
sin(12t/π) = -1
12t/π = 3π/2
t = (3π/2)(12/π) = 18
so the max occurs at 6 units of time and the min at 18 units of time
period = 24 units of time
As it stands with no phase shift,
at t = 0 , temp = 27
at t = 6, temp = 32.9
at t = 12 , temp = 27
at t =18, temp = 21.1
at t = 24 , temp = 27 , etc
To have a phase shift, there must have been something given such as,
we know that if t = 4, temp = some value
Well I have a chart so I know that when t = 4 temp = 21.1
So where would I go from there?
Ignore most of the above, I have some of the fractions upside down,
the period is right at 24, making k = π/12
You must have meant:
k = π/12 , not period = π/12
and our basic equation is
T(t) = 5.9 sin ((π/12)t -c ) + 27
Now that you told me that temp = 21.1 when t = 4 , we can say
21.1 = 5.9 sin((π/12)(4) - c) + 27
sin (π/3 - c) = -1
π/3 - c =3π/2
-c = 3π/2 - π/3 = 7π/6
T(t) = 5.9 sin( πt/12 + 7π/6 ) + 27
testing:
if t = 16 , we should get 32.9
T(16) = 5.9 sin(16π/12 + 7π/6 ) + 27
= 5.9 sin (5π/2) + 27
= 5.9(1) + 27 = 32.9 ----- YEAHHHH
Thank you so much you've been a HUGE help, this question has had me stumped for so long.
To find the phase shift of a sine function, you need to determine the horizontal translation of the graph, which indicates how much the graph is shifted to the left or right. In the equation T(t) = a sin(kt - c) + d, the value of c represents this phase shift.
To find the phase shift, you can use the information given about the graph. The temperature function T(t) oscillates between a maximum of 32.9 and a minimum of 21.1. The midpoint between these two values is the average of the maximum and minimum, which is (32.9 + 21.1)/2 = 27.
Since the general form of the sine function is T(t) = a sin(kt - c) + d, the value of d is the vertical translation of the graph, which is the midpoint value of 27 in this case.
Now, we need to determine the phase shift in terms of the period of the function. The period of a sine function is the horizontal distance between two consecutive maximum or minimum values. In this case, you have mentioned that the period is pi/12, which means that the graph repeats itself every pi/12 units of time.
To find the phase shift, you can calculate the difference between the given time t and the time at which the graph crosses the x-axis (the time at which the argument of the sine function, kt - c, is equal to zero). Let's call this time t0.
The phase shift, in terms of the period, is given by the formula:
Phase shift = (t - t0) / (1/period)
Now, to calculate t0, you need to find the time at which the temperature function crosses the x-axis. This occurs when the argument of the sine function, kt - c, is equal to zero. Since the amplitude is positive and the maximum and minimum temperatures occur at the highest and lowest points of the graph, respectively, we can conclude that the graph starts at its maximum.
Therefore, the phase shift, c, can be found by solving the equation kt - c = 0, where t is the time at which the graph starts at its maximum. Since this is a sine function, the moment it starts at its maximum corresponds to a phase shift of zero (no horizontal translation), so the equation becomes kt = 0. Taking into account the given period of pi/12, we can set k * t = 0 * (pi/12) = 0, implying t = 0.
Now that we have t0 = 0, the formula for the phase shift simplifies to:
Phase shift = (t - t0) / (1/period)
= t / (1/period)
= t * period
Substituting the given period of pi/12, the phase shift can be calculated as:
Phase shift = t * (pi/12)
In this case, the phase shift, c, is the quantity we're trying to find.