The temperature in your house is controlled by a thermostat. The temperatures will vary according to the sinusoidal function:f(x)=6sin(pi/12(x-11)+19, where f(x) represents the temperature in degrees Celsius (°C) and x is hours since midnight. What is the temperature of your house at 3:00 p.m.?
I got 19 celcius
And its maximum temperature, I got 25 celcius.
3:00 pm is 15 hours after midnight
6sin(pi/12 (15-11))+19
6sin(pi/12 * 4)+19
6sin(pi/3)+19
6*0.866+19
24.196
6+19=25 is the maximum possible temperature
To find the temperature at 3:00 p.m., we need to convert the time to hours since midnight. Since 3:00 p.m. is 15:00, we subtract 12 to get the number of hours since midnight.
x = 15 - 12 = 3
Now we can substitute this value into the sinusoidal function:
f(x) = 6sin(pi/12(x-11)) + 19
f(3) = 6sin(pi/12(3-11)) + 19
f(3) = 6sin(pi/12(-8)) + 19
f(3) = 6sin(-2pi/3) + 19
Using the unit circle or a calculator, we find that sin(-2pi/3) = -√3 / 2.
f(3) = 6(-√3 / 2) + 19
f(3) = -3√3 + 19
So, the temperature in the house at 3:00 p.m. is approximately -3√3 + 19 degrees Celsius.
To find the temperature of your house at 3:00 p.m., you need to substitute the value of x for 15 in the given function f(x) = 6sin((π/12)(x-11)) + 19.
Let's break down the process step by step:
Step 1: Convert 3:00 p.m. to hours since midnight.
To do this, you need to calculate the number of hours between midnight and 3:00 p.m.
Since there are 12 hours on a clock, from midnight to noon, we have 12 hours. Therefore, from noon to 3:00 p.m., there are an additional 3 hours. This gives us a total of 12 + 3 = 15 hours.
Step 2: Substitute the value of x into the function.
Now that we know x = 15, we can substitute it into the equation:
f(15) = 6sin((π/12)(15-11)) + 19
Step 3: Simplify the equation.
Let's calculate the inner part first:
(π/12)(15-11) = (π/12)(4) = (π/3)
Substituting the simplified value, we get:
f(15) = 6sin((π/3)) + 19
Since sin(π/3) = 0.866, we can substitute this value into the equation:
f(15) = 6(0.866) + 19
Step 4: Calculate the final result.
Now, all that's left is to perform the calculations:
f(15) = 6(0.866) + 19
f(15) ≈ 5.196 +19
f(15) ≈ 24.196
Therefore, the approximate temperature of your house at 3:00 p.m. is 24.196°C.