I have a question I do not think anybody answered it
Researches find a creature form an alienplanet. Its body temperature is varying sinusoidally with time. 35 min after they start timing, it reaches a high of 120 F. 20 min after that it reaches its next low, 104 F.
How do I come up with this equation I'm getting this
y = 8 sin (pi X)/20
I do not know if there is a phase sift in this equation I'm obviously doing something worng please tell me how to come up with this equation thank you
you are right on the amplitude 8.
Now half of period is 20 min. So period is 40 min, it needs a 35/40 *PI shift.
check that.
how do we now that the phase shift is (35pi)/40
please explain
THANKS
Lets do it again:
the average temp is 112
temp=112+8sin(wt+theta)
now, you know a zero is between 10 min before the high. Which is 25 min after start.
Theta= 25*2PI/40=3.93 radians
temp= 112+ 8 sin(wt-3.93)
and w= 2PI/40
check
temp at 35:
temp=112+8sin(2PI/40 *35-3.93)=120
I hope that helps.
To determine the equation that represents the creature's body temperature varying sinusoidally with time, we can use the given information.
Let's analyze the key details stated in the question:
1. The creature reaches a high of 120°F, 35 minutes after they start timing.
2. 20 minutes after reaching the high, the creature reaches its next low, which is at 104°F.
To create an equation that captures these temperature changes, we need to consider the amplitude, period, and phase shift.
1. Amplitude: The amplitude is half the difference between the high and low points. In this case, it would be [(120 - 104) / 2] = 8°F.
2. Period: The period is the time it takes for the creature to complete one full cycle, in this case, from one high point to another high point or from one low point to another low point. From the given information, we know that it takes 35 minutes to reach the high and another 20 minutes to reach the next low, which makes it a total of 35 + 20 = 55 minutes for one complete cycle. The period can be determined by converting 55 minutes into radians - since one complete cycle is 2π radians - by the formula: Period (in radians) = (time taken for one cycle / total time for one complete cycle) * 2π. Substituting the values, we get the period as: Period (in radians) = (55 / 55) * 2π, which simplifies to 2π.
3. Phase Shift: The phase shift represents the horizontal shift of the wave. In this case, we are not given any information about a phase shift, so we can assume there is no phase shift.
Putting all of this information together, the general equation for a sinusoidal function is:
y = A sin(B(x - C)) + D,
where:
- A represents the amplitude,
- B represents the value determining the period (in this case, 2π),
- C represents the phase shift,
- D represents any vertical shift.
In our case, since we determined that there is no phase shift, the equation becomes:
y = 8 sin(2πx) + D.
Now, we need to determine the vertical shift, D. To do that, substitute either of the given temperature points into the equation:
120 = 8 sin(2π(35)) + D,
104 = 8 sin(2π(55)) + D.
Solving these two equations will give us the value of D. Let's solve them:
120 = 8 sin(70π) + D,
104 = 8 sin(110π) + D.
By solving these, we get D ≈ 112.73.
Now, substituting this value back into the equation, we have:
y = 8 sin(2πx) + 112.73,
which is the equation that represents the creature's body temperature varying sinusoidally with time.