Record, indicate that t hours past midnight, the temperature at the local airport was T(t)=-0.3t^2+4t+10degree celcius. What was average temperature in farenheit at airport between 9am and noon?
Plz damon show step
integral T dt
= .1 t^3 + 2 t^2 + 10 t
at t = 12
= .1(1728) + 2(144) + 120
at t = 9
= .1(729) + 2(81) + 90
subtract value at 9 from value at 12
divide by (12-9)
hmmm just look for the antiderivative for dt
That is correct. The average value of a function over an interval if
∫[a,b] f(t) dt
------------------
b-a
To find the average temperature in Fahrenheit at the airport between 9 am and noon, we first need to determine the temperature at these specific times.
Given that:
- t represents the number of hours past midnight,
- T(t) represents the temperature in degrees Celsius at time t, and
- we want to find the average temperature between 9 am and noon,
Let's start by converting the given temperature function from degrees Celsius to Fahrenheit.
The formula to convert degrees Celsius (C) to degrees Fahrenheit (F) is: F = (C * 9/5) + 32.
Now let's substitute T(t) into the formula above and solve for the temperature in Fahrenheit at time t:
F(t) = (T(t) * 9/5) + 32
For our specific times, we need to find F(9) and F(12).
To find F(9):
- Substitute t = 9 into the original temperature function T(t): T(9) = -0.3(9)^2 + 4(9) + 10.
- Calculate T(9) using the order of operations: T(9) = -0.3(81) + 36 + 10 = -24.3 + 36 + 10 = 21.7.
- Substitute T(9) into the Fahrenheit conversion formula: F(9) = (21.7 * 9/5) + 32 = 39.06 + 32 = 71.06°F.
To find F(12):
- Substitute t = 12 into the original temperature function T(t): T(12) = -0.3(12)^2 + 4(12) + 10.
- Calculate T(12) using the order of operations: T(12) = -0.3(144) + 48 + 10 = -43.2 + 48 + 10 = 14.8.
- Substitute T(12) into the Fahrenheit conversion formula: F(12) = (14.8 * 9/5) + 32 = 26.64 + 32 = 58.64°F.
Now that we have F(9) and F(12), we can calculate the average temperature between 9 am and noon.
Average temperature = (F(9) + F(12)) / 2
= (71.06 + 58.64) / 2
= 129.7 / 2
= 64.85°F
Therefore, the average temperature in Fahrenheit at the airport between 9 am and noon is approximately 64.85°F.