Recorded data indicates that on a recent date t hours past midnight, the temperature at Logan airport was
f(t) = 14sin(1/3t-2) + 60
degrees Fahrenheit. What was the average temperature between 9 am and noon?
f(t) = 14sin(t/3-2) + 60
∫f(t)dt from 9 to 12
=[ 60*t-42*cos(t/3-2) ] from 9 to 12
=220.17
Average temperature over 3 hours
= 220.17/3
= 73.4°
Note to Aviana:
in the absence of parentheses, multiplications and divisions are performed before additions and subtractions.
Even when 14sin(1/3t-2) + 60 is interpreted correctly as 4sin(t/3-2) + 60, it is a good idea to insert parentheses to emphasize the intention, such as : 4sin((1/3)t-2) + 60
To find the average temperature between 9 am and noon, we need to find the average value of the temperature function f(t) over that time interval.
First, let's convert the time from hours to the corresponding values of t in the temperature function. Since 9 am is 3 hours past midnight, and noon is 12 pm, which is 12 hours past midnight, we can substitute these values into the temperature function:
f(3) = 14sin(1/3(3)-2) + 60
f(12) = 14sin(1/3(12)-2) + 60
Now, evaluate these expressions:
f(3) = 14sin(1/3-2) + 60 = 14sin(-5/3) + 60
f(12) = 14sin(4-2) + 60 = 14sin(2) + 60
To find the average temperature, we need to find the integral of the temperature function over the interval [3, 12] and then divide it by the length of the interval (12 - 3 = 9 hours):
Average temperature = (1/9) * ∫[3,12] (14sin(1/3t-2) + 60) dt
To solve this integral, first integrate each term separately:
(1/9) * [ -42cos(1/3t - 2) + 60t ] evaluated from 3 to 12
Evaluate the integral at the upper and lower limits:
(1/9) * [ -42cos(1/3(12) - 2) + 60(12) - (-42cos(1/3(3) - 2) + 60(3)) ]
Simplifying the expression:
(1/9) * [ -42cos(2) + 720 + 42cos(-5/3) - 180 ]
Evaluate the cosine terms:
cos(2) ≈ -0.4161
cos(-5/3) ≈ -0.8716
Substituting these values into the equation:
(1/9) * [ -42(-0.4161) + 720 + 42(-0.8716) - 180 ]
Calculate the expression:
Approximately,
Average temperature ≈ 88.724 degrees Fahrenheit
Therefore, the average temperature between 9 am and noon at Logan airport, based on the given temperature function, is approximately 88.724 degrees Fahrenheit.