Ice cream drips out of the bottom of an ice cream cone on a hot day at a rate of r(t) mL per second, as a child eats it slowly, where t is in seconds. If r(t) = 10 e^{-k t}, complete the definite integral expressing the quantity of ice cream lost in the first 3 minutes(s). (Watch units!) Don't calculate anything.
Total quantity = \int_{a}^{b} 10 e^{-kt} dt
(where a = and b = ).
What are the units of the answer to this integral? Write as you would on a written assignment.
To complete the definite integral expressing the quantity of ice cream lost in the first 3 minutes, we need to integrate the function r(t) = 10e^(-kt) with respect to t from the initial time (a) to final time (b).
The initial time can be calculated using the number of seconds in 3 minutes:
a = 0 seconds
The final time can be calculated using the number of seconds in 3 minutes:
b = 3 minutes * 60 seconds/minute = 180 seconds
Thus, the definite integral to represent the total quantity of ice cream lost in the first 3 minutes is:
Total quantity = ∫[0 to 180] 10e^(-kt) dt
Now, let's discuss the units of the answer to this integral.
The integral represents the accumulation of the rate of ice cream lost over a specific time interval. The rate r(t) is given in milliliters per second (mL/s), and the time t is in seconds (s).
Therefore, the units of the integral should be (mL/s) * s, which simplifies to mL. So, the units of the answer to this integral would be milliliters (mL).