Respuesta :
Answer:
The model of the F-sharp note is [tex]f(x) = sin(123.75 \pi)[/tex]
The model of the temperature is [tex]T(t)= 20 sin (\frac{\pi}{2}(t+15) ) +50[/tex]
Step-by-step explanation:
From the question we are told that
The frequency is [tex]f = 61.875\ Hz[/tex]
The Amplitude is A = 1
The phase shift is [tex]\theta = 0[/tex]
The angular velocity of a wave can be given as
[tex]w = 2 \pi f[/tex]
Substituting values
[tex]w = 2 \pi *61.875[/tex]
[tex]= 123.75\pi[/tex]
Now the sine function model of this note is given as
[tex]f(x) = A sin (wt +\theta)[/tex]
Substituting values
[tex]f(x) = sin(123.75 \pi)[/tex]
The minimum temperature is [tex]T_{min} = 30^o[/tex]
The maximum temperature is [tex]T_{max} = 70^o[/tex]
The average temperature is [tex]T_{avg} = \frac{70+30 }{2} = 50^o[/tex]
The Amplitude is [tex]A = \frac{T_{max} - T_ [min}{2} = \frac{70-30}{2} =20^0[/tex]
The period = 24 This because the function is modeled in such a way that 24 hour is time to complete a full cycle
The General sin equation is mathematically given as
[tex]T(t) = A sin (b (t - c )) +d[/tex]
Where A is the Amplitude = 20
b is the angular speed
c is the phase shift
d is the vertical transition
And T(t) is the temperature after time t
And the period is evaluates as [tex]T = \frac{2 \pi}{b}[/tex] since [tex]T = \frac{1}{frequency}[/tex]
This implies that [tex]b= \frac{2 \pi}{T} = \frac{2 \pi}{24 }=\frac{\pi}{12}[/tex]
Since average is at 50 then it means that the wave graph has been shifted by 50 unites
d = 50
Now from our question
t = 3 hours when T(t) = 30
This means that
[tex]T(3) =20 sin (\frac{\pi}{12}(3 -C) )+50[/tex]
=> [tex]30 = 20 sin (\frac{\pi}{12} (3-c)) +50[/tex]
=> [tex]20 sin (\frac{\pi}{12} (3-c)) = 30 -50= -20[/tex]
=> [tex]sin (\frac{\pi}{12} (3 -c )) = -1[/tex]
Since [tex]-1 = sin \frac{3 \pi}{2}[/tex] we have
[tex]sin (\frac{\pi}{12} (3 -c )) = sin(\frac{3 \pi}{2} )[/tex]
=> [tex]3-c = \frac{sin(\frac{\pi}{12} )}{sin (\frac{3 \pi}{2} )}[/tex]
=> [tex]-c = 18 -3[/tex]
=> [tex]c = - 15[/tex]
Therefore the sinusoidal model of the temperature is
[tex]T(t) = 20 sin (\frac{\pi}{12} (t - (-15)) ) +50[/tex]
[tex]T(t)= 20 sin (\frac{\pi}{2}(t+15) ) +50[/tex]
Answer:
a) [tex]g(x) = sin (123.75\pi t)[/tex]
b) [tex]T(t) = 20 sin (\frac{\pi }{12} (t+15))+50[/tex]
Step-by-step explanation:
Frequency, f = 61.875 Hz
[tex]g(x) = A sin (wt + \alpha)[/tex]...........(1)
Amplitude, A = 1
Since there is no phase shift, [tex]\alpha = 0[/tex]
[tex]w = 2 \pi f[/tex]
[tex]w = 2\pi * 61.875[/tex]
[tex]w = 123.75 rad/s[/tex]
Substituting the values of A and w into equation (1)
[tex]g(x) = sin (123.75\pi t)[/tex]
b)
The general equation for a sinusoid
[tex]T(t) = A sin (B(t-C))+D[/tex].................(2)
Lot temperature = 30⁰
High temperature = 70⁰
Amplitude, [tex]A = \frac{70-30}{2} \\[/tex]
A = 20⁰
Vertical shift, [tex]D = \frac{70 + 30}{2}[/tex]
D = 50⁰
Since the low temperature of 30⁰ occurs at 3 AM
t = 3, T(t) = 30
Period, [tex]T = \frac{2\pi }{B}[/tex], [tex]T = 24 hrs[/tex]
[tex]B = \frac{2\pi }{T}[/tex]
[tex]B = \frac{\pi }{12}[/tex]
[tex]30 = 20 sin (\frac{\pi }{12} (3-C))+50[/tex]
[tex]-20 = 20 sin (\frac{\pi }{12} (3-C))[/tex]
[tex]-1 = sin (\frac{\pi }{12} (3-C))[/tex]
[tex]sin^{-1} (-1)= (\frac{\pi }{12} (3-C))[/tex]
[tex]3\pi/2 = (\frac{\pi }{12} (3-C))\\18 = 3 - C\\C = -15[/tex]
Substituting the values of A, B, C, D into equation (2)
[tex]T(t) = 20 sin (\frac{\pi }{12} (t+15))+50[/tex]
