Questions: Suppose your monthly revenue from selling used iPads is R(t) = 290 + 13t (0 ≤ t ≤ 4) dollars per month, where t represents months from the beginning of the year, while your monthly costs are C(t) = 105 + 2t (0 ≤ t ≤ 4) dollars per month. Find the total accumulated profit for 0 ≤ t ≤ 4. (Round your answer to a whole number.)

Solution Steps

To find the total accumulated profit over the given time period, we need to calculate the profit for each month and then sum these profits. The profit for each month is the revenue minus the cost. We will integrate the profit function, which is the difference between the revenue function \( R(t) \) and the cost function \( C(t) \), over the interval from \( t = 0 \) to \( t = 4 \).

The monthly revenue from selling used iPads is given by the function: \[ R(t) = 290 + 13t \] The monthly costs are represented by the function: \[ C(t) = 105 + 2t \]

The profit function \( P(t) \) is defined as the difference between revenue and costs: \[ P(t) = R(t) - C(t) = (290 + 13t) - (105 + 2t) = 11t + 185 \]

To find the total accumulated profit over the interval \( 0 \leq t \leq 4 \), we integrate the profit function: \[ \text{Total Profit} = \int_0^4 P(t) \, dt = \int_0^4 (11t + 185) \, dt \] Calculating this integral gives: \[ \text{Total Profit} = 828 \]

Final Answer