Records indicate that t hours past midnight, the temperature at the local airport was $f ( t ) = - 0.3 t ^ { 2 } + 6 t + 12$ degrees Fahrenheit. What was the average temperature at the airport between 5:00 A.M. and noon? Round your answer to one decimal place, if necessary.
A) 75.1 degrees F
B) 40.1 degrees F
C) 28.1 degrees F
D) 16.5 degrees F

Question

Quizplus · Accepted Answer

To find the average temperature between 5:00 A.M. and noon (7 hours later), we integrate the temperature function $f(t) = -0.3t^2 + 6t + 12$ from $t = 5$ to $t = 12$ and then divide by the interval length (7 hours). The integral of $f(t)$ from 5 to 12 gives the total temperature over the period, and dividing by 7 gives the average temperature.$$\text{Average temperature} = \frac{1}{12-5} \int_{5}^{12} (-0.3t^2 + 6t + 12) dt$$$$= \frac{1}{7} \left[ -0.1t^3 + 3t^2 + 12t \right]_{5}^{12}$$$$= \frac{1}{7} \left( (-0.1 \cdot 12^3 + 3 \cdot 12^2 + 12 \cdot 12) - (-0.1 \cdot 5^3 + 3 \cdot 5^2 + 12 \cdot 5) \right)$$$$= \frac{1}{7} \left( (-0.1 \cdot 1728 + 3 \cdot 144 + 144) - (-0.1 \cdot 125 + 3 \cdot 25 + 60) \right)$$$$= \frac{1}{7} \left( (-172.8 + 432 + 144) - (-12.5 + 75 + 60) \right)$$$$= \frac{1}{7} \left( 403.5 - 122.5 \right)$$$$= \frac{1}{7} \cdot 281 = 40.1$$Therefore, the average temperature at the airport between 5:00 A.M. and noon is 40.1 degrees Fahrenheit.

Records Indicate That T Hours Past Midnight, the Temperature at the Local