The temperature in your house is controlled by a thermostat. The temperatures will vary according to the sinusoidal function:
f(x)=7sin(π/12(x-11))+33 where f(x) represents the temperature in degrees Celsius (°C) and x is hours since midnight. What is the temperature of your house at midnight?
a) 7°C
b) 16°C
c) 22°C
d) 31°C
I tried plugging in zero for x since no time had passed since midnight, but my result wasn't one of the answer choices. Thanks!!
hmmm
think it means
t = 33 + 7 sin [ (pi/12)(x-11) ]
when x = 0
we have sin [-11 pi/12]
which is - sin [11*180/12] = -sin(165 degrees)
= -.259
so
t = 33 - 7(.259) = 31.2 degrees F
To find the temperature of your house at midnight, we need to evaluate the function f(x) when x is equal to zero. Let's substitute x = 0 into the given equation:
f(x) = 7sin(π/12(x-11)) + 33
f(0) = 7sin(π/12(0-11)) + 33
f(0) = 7sin(π/12(-11)) + 33
Now, let's calculate the value of sin(π/12(-11)). Since sin(-θ) = -sin(θ), we have:
f(0) = 7(-sin(π/12(11))) + 33
Now, let's calculate the value of sin(π/12(11)).
The sine function has a periodicity of 2π, which means sin(2π) = sin(0).
Therefore, sin(π/12(11)) = sin(2π + π/12(11)), since adding 2π doesn't change the sine value.
Now, let's simplify: π/12(11) = π/12 * 11 = π/12 * 12/1 = π/1 = π.
So, sin(π/12(11)) = sin(2π + π) = sin(3π).
Since the sine function is periodic every 2π, sin(3π) is equal to sin(π), as the value of 3π is equivalent to π after one full period.
Now, sin(π) = 0.
Therefore, sin(π/12(11)) = 0.
Substituting this value back into the equation:
f(0) = 7(-sin(π/12(11))) + 33
f(0) = 7(0) + 33
f(0) = 0 + 33
f(0) = 33
Hence, the temperature of your house at midnight is 33°C.
Therefore, none of the given answer choices (a) 7°C, (b) 16°C, (c) 22°C, or (d) 31°C are correct.