A company estimates that its sales will grow continuously at a rate given by the function \( S^{\prime}(t)=12 e^{t} \) where \( S^{\prime}(t) \) is the rate at which sales are increasing, in dollars per day, on day \( t \). a) Find the accumulated sales for the first 7 days. b) Find the sales from the 2 .

To solve the problem, we need to find the accumulated sales over the first 7 days and the sales from day 2 to day 7. ### Part (a): Accumulated Sales for the First 7 Days The rate of change of sales is given by: \[ S'(t) = 12 e^t \] To find the accumulated sales \( S(t) \) over the first 7 days, we need to integrate \( S'(t) \) from \( t = 0 \) to \( t = 7 \): \[ S(t) = \int S'(t) \, dt = \int 12 e^t \, dt \] Calculating the integral: \[ S(t) = 12 e^t + C \] Since we are interested in the accumulated sales from day 0 to day 7, we can evaluate the definite integral: \[ \text{Accumulated Sales from } 0 \text{ to } 7 = S(7) - S(0) \] Calculating \( S(7) \) and \( S(0) \): \[ S(7) = 12 e^7 + C \] \[ S(0) = 12 e^0 + C = 12 + C \] Now, substituting these into the accumulated sales formula: \[ \text{Accumulated Sales} = (12 e^7 + C) - (12 + C) = 12 e^7 - 12 \] Thus, the accumulated sales for the first 7 days is: \[ \text{Accumulated Sales} = 12 (e^7 - 1) \] ### Part (b): Sales from Day 2 to Day 7 To find the sales from day 2 to day 7, we need to evaluate the definite integral of \( S'(t) \) from \( t = 2 \) to \( t = 7 \): \[ \text{Sales from Day 2 to Day 7} = \int_{2}^{7} S'(t) \, dt = \int_{2}^{7} 12 e^t \, dt \] Calculating the integral: \[ \int 12 e^t \, dt = 12 e^t + C \] Now we evaluate the definite integral: \[ \text{Sales from Day 2 to Day 7} = \left[ 12 e^t \right]_{2}^{7} = 12 e^7 - 12 e^2 \] Thus, the sales from day 2 to day 7 is: \[ \text{Sales from Day 2 to Day 7} = 12 (e^7 - e^2) \] ### Summary of Results - **Accumulated Sales for the First 7 Days**: \( 12 (e^7 - 1) \) - **Sales from Day 2 to Day 7**: \( 12 (e^7 - e^2) \)