A group of scientists found a new species of spider in the desert. The body temperature of the spider appears to vary sinusoidal over time. A maximum body temperature of the spider reaches 125? after 15 minutes from the start of the examination. Then, 14 minutes later, the body temperature falls to a minimum of 99?. The scientists would like to write an equation to model the body temperature of the spider over time. Which function will model their findings?
I got f(t)=13sin (pi/7(x-1)+122
amplitude is (125-99)/2 = 13
The center line is 125-13 = 112, not 122
2pi/k = 2*14, so k = pi/14
Since f(15) = 125, the max, that means that f(1) = 99, the minimum
cosx has a max at x=0, so -cos(x-1) has a minimum at x=1.
f(t) = 13 cos(pi/14 (x-1)) + 112
The graph at
http://www.wolframalpha.com/input/?i=-cos%28pi%2F14+%28x-1%29%29
shows that the cosine part has its first max at x=15 and its next min at x=29.
You left t out !
max at 15 min
min at 15+14 = 29
so
14 min from sin = +1 to sin = -1
that is half a period so T = 28 min
2pi t/T = 2 pi at t=T we disagree about all this so pi t/14 is 2 pi when t = T = 28
range from max to min = 2 A = 125-99 = 26
so
A = 13 (check with you :)
average = (125+99)/2 = 224/2 = 112 (check with you)
so so far
f(t) = 112 + 13 sin (pi t/14 - phase)
when t = 15, f(15) = 125
125 = 112 + 13 sin (pi*15/14 - phase)
sin (pi*15/14 - phase) = 1
so
(15/14) pi - phase = pi/2
phase = pi (8/14)
so
f(t) = 112 + 13 sin (pi t/14 - 8 pi/14)
or
f(t) = 112 + 13 sin [ (pi/14)(t-8) ]
The correct equation to model the body temperature of the spider over time is:
f(t) = 13 sin((π/7)(t - 15)) + 122
This equation represents a sinusoidal function, where t represents time (in minutes) and f(t) represents the body temperature of the spider. In the equation, the maximum body temperature is 125°F, so we shift the sin function vertically by adding 122. The amplitude of the sinusoidal function is 13, which indicates the difference between the maximum and minimum values of the temperature. Finally, the period of the function is 14 minutes, as it takes 14 minutes for the temperature to go from the maximum to the minimum value.
To model the findings of the scientists, you can use a sinusoidal function that represents the body temperature of the spider over time.
Let's break down the given information to find the appropriate function:
1. The body temperature of the spider varies sinusoidally over time.
2. The maximum body temperature reaches 125? after 15 minutes from the start.
3. 14 minutes later, the body temperature falls to a minimum of 99?.
To find the function, we need to identify the key features of a sinusoidal function: amplitude, period, phase shift, and vertical shift.
Amplitude represents the vertical distance from the midline to the maximum or minimum point of the function. In this case, the amplitude is half the difference between the maximum and minimum temperatures:
Amplitude (A) = (125 - 99) / 2 = 13.
The period represents the length of one complete cycle of the sinusoidal function. Here, one full cycle seems to take place in 29 minutes, which is the sum of the time between the maximum and minimum points:
Period (P) = 15 + 14 = 29.
The phase shift indicates any horizontal displacement of the graph. In this case, the sinusoidal function seems to start at the 15th minute, which corresponds to a phase shift of -15.
Finally, the vertical shift refers to any vertical translation of the graph. The range of temperatures observed is from 99? to 125?, so the vertical shift is (99 + 125) / 2 = 112.
Now we can write the equation using the information above:
f(t) = A * sin[(2π / P) * (t - phase shift)] + vertical shift
Substituting the known values:
f(t) = 13 * sin[(2π / 29) * (t - 15)] + 112.
Therefore, the function that models the body temperature of the spider over time is:
f(t) = 13 * sin[(2π / 29) * (t - 15)] + 112.