The function $k(t) = 8 −\operatorname{cos} t + 0.01t^2 + 0.01t$ helped Kira determine the rate at which electricity was being used on any day $t$ of the month. However, to manage total electrical production, Kira needs to determine a function that will accumulate the energy used over the course of the month. That is, she needs a function $K(t)$ that will determine the total energy used from the beginning of the month to time $t$.

1. Find a function $K(t)$ that will accumulate the power used over time $t$. Be sure that $K(0) = 0$. What is its relationship to $k(t)$?

   Hint (a):

   Notice that lower-case, $k(t)$, is a rate function.

2. Use $K(t)$ to find the total electrical energy used during this month.

   Hint (b):

   Assume the month has $30$ days.

   More Help (b):

   'Total electrical consumption' is another way to say 'accumulated power.'

3. Sketch a graph of $K(t)$ for this $30$-day period.

   Answer (c):

   Make sure that this graph agrees with your function for $K(t)$.
   

4. Calculate the average power used in kilowatts for this month. Did your answer match that in problem 6-151? How is this rate represented geometrically in the graph?