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, 28 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?
a.) f(t)=13sin(2pi/28)*(t-1)+112
b.) f(t)=13sin(pi/28)*(t-1)+112
c.) f(t)=13sin(pi/56)*(t-1)+112
d.) f(t)=26sin(pi*t/28)+112
How do I calculate Amplitude, period, etc. to find a function? I think that the correct answer is c, but I'm not positive. Thanks!
amplitude is half the distance between the extremes. In this case, (125-99)/2 = 13
The midline (where sin(x) = 0) is (125+99)/2 = 112
The period is twice the time from max to min, or 28*2 = 56
So, we're looking at something like
y=13 sin(pi/28 t)+112
Now for the phase shift. The max was at t=15, so y=0 at t=1, 1/4 period earlier.
y=13 sin(pi/28 (t-1))+112
that is, choice (b)
see the graph at
http://www.wolframalpha.com/input/?i=%3Dsin%28%28pi%2F28%29*%28x-1%29%29
It goes from 125 to 99 in 28 minutes, so a whole cycle or a period would be 56 minutes
period = 2π/k = 56
k = 2π/56 = π/28
which automatically rules out a) and c)
The sin(anything) has a max of 1 and a min of -1
so 13+112 = 125
-13+112 = 99
So a), b), and c) have that property, ruling out d)
(unless none of them are correct, b) is it )
let's test it for the given values:
if t = 15, we should get 125
temp = 13sin(π/28)(15-1) + 112
= 13 sin (14π/28) + 112 = 13(1) + 112 = 125 , ok!
if t = 43, we should get 99
temp = 13sin (π/28)(43-1) + 112
= 13 sin (3π/2) + 112 = 13(-1) + 112
= 99, OK!!
so b)
Thanks SO much!!
To find the function that models the body temperature of the spider over time, you need to analyze the given information to calculate the amplitude, period, and other parameters.
Amplitude:
The amplitude of a sinusoidal function is the maximum variation from the average value. In this case, the maximum body temperature is given as 125°, and the minimum temperature is 99°. The difference between these two values is 125° - 99° = 26°, which represents the amplitude.
Period:
The period of a sinusoidal function is the length of one complete cycle. In this case, the maximum body temperature takes 15 minutes after the start, and then 28 minutes later, it reaches the minimum temperature. Therefore, the complete cycle takes a total of 28 minutes.
Phase Shift:
The phase shift represents any horizontal shift in the sinusoidal function. In this case, there is no explicit information about a phase shift, so we assume it starts at t=0. This means there is no phase shift.
Calculating the Function:
Now that we have found the amplitude, period, and phase shift, we can use the general form of the sinusoidal function, which is:
f(t) = A*sin(B*(t - C)) + D
Using the given values, the function that models the spider's body temperature over time is:
f(t) = 26*sin((pi/56)*(t - 0)) + 112
This simplifies to:
f(t) = 26*sin((pi/56)*t) + 112
Therefore, the correct answer is option c.) f(t) = 13sin(pi/56)*(t-1)+112.
Make sure to double-check your calculations and provide additional context if you suspect another option might be correct.