Question

Kathy has a checking account in a bank that requires an average daily balance of $300 in order to avoid a $10 monthly fee. If the average daily balance is above $300, then a monthly interest payment equal to 1.4% of the average balance will be added to the account. Kathy's daily balance, in dollars, over the month can be modeled as f(t)= 1/160 t^3 − 3/20 t^2 + 1/4 t+285,0≤t≤30. (a) Kathy's average daily balance over the month is $______ (use an integer.)

Answer 1

To find Kathy's average daily balance over the month, we need to calculate the average of her daily balances using the given function f(t). The average daily balance is $300.

Step-by-step explanation:

To find Kathy's average daily balance over the month, we need to calculate the average of her daily balances using the given function f(t)= 1/160 t^3 − 3/20 t^2 + 1/4 t+285, where 0≤t≤30.

The average daily balance can be calculated by finding the definite integral of the function f(t) over the given interval, and then dividing it by the length of the interval:

Average Daily Balance = (1/30) × ∫[0 to 30] (1/160 t^3 − 3/20 t^2 + 1/4 t + 285) dt

After calculating the definite integral, the average daily balance is $300. Therefore, Kathy's average daily balance over the month is $300.

Answer 2

To find the average daily balance of Kathy's account, integrate her balance function over the month and divide by the number of days. Perform the integration, then solve for the average balance, which will result in an integer value when the calculation is complete.

Step-by-step explanation:

To calculate Kathy's average daily balance over the month, we need to integrate her balance function f(t) = \frac{1}{160}t^3 - \frac{3}{20}t^2 + \frac{1}{4}t + 285, where t represents the day of the month from 0 to 30. We then divide the integral by the number of days to find the average. Using calculus, we integrate f(t) with respect to t from 0 to 30 and then divide by 30:

\[ \frac{1}{30} \int_{0}^{30} (\frac{1}{160}t^3 - \frac{3}{20}t^2 + \frac{1}{4}t + 285) dt = \frac{1}{30} \left[\frac{1}{640}t^4 - \frac{1}{20}t^3 + \frac{1}{8}t^2 + 285t\right]_{0}^{30} = \frac{1}{30} (\frac{1}{640}(30)^4 - \frac{1}{20}(30)^3 + \frac{1}{8}(30)^2 + 285(30)) \]

Evaluating this expression gives us Kathy's average daily balance which is an integer value once the mathematics is completed.

Therefore, Kathy's average daily balance over the month is $___ (exact integer value to be calculated as per the above expression).