The lotal sales of a company (in millions of dollars) I months from now are given by \( \mathrm{S}(\mathrm{t})=0.051^{3}+0.4 \mathrm{t}^{2}+21+9 \). (A) Find \( \mathrm{S}^{\prime}(t) \). (B) Find \( \mathrm{S}(2) \) and \( \mathrm{S}^{\prime}(2) \) (to two decimal places). (C) Interpret \( \mathrm{S}(10)=119.00 \) and \( \mathrm{S}^{\prime}(10)=25.00 \). (A) \( \mathrm{S}^{\prime}(t)=0.15 t^{2}+0.8 t+2 \) (B) \( \mathrm{S}^{(2)}=\square \) \( \mathrm{S}^{\prime}(2)=4.20 \)

Let's solve each part of the problem step by step. ### Given: The sales function is: \[ \mathrm{S}(\mathrm{t}) = 0.05 \mathrm{t}^{3} + 0.4 \mathrm{t}^{2} + 2\mathrm{t} + 9 \] where \( \mathrm{S}(\mathrm{t}) \) is in millions of dollars and \( \mathrm{t} \) is the time in months. ### (A) Find \( \mathrm{S}^{\prime}(t) \). To find the first derivative \( \mathrm{S}^{\prime}(t) \), differentiate \( \mathrm{S}(t) \) with respect to \( t \): \[ \mathrm{S}^{\prime}(t) = \frac{d}{dt}\left(0.05 \mathrm{t}^{3} + 0.4 \mathrm{t}^{2} + 2\mathrm{t} + 9\right) = 0.15 \mathrm{t}^{2} + 0.8 \mathrm{t} + 2 \] **Answer:** \[ \mathrm{S}^{\prime}(t) = 0.15 \mathrm{t}^{2} + 0.8 \mathrm{t} + 2 \] ### (B) Find \( \mathrm{S}(2) \) and \( \mathrm{S}^{\prime}(2) \) (to two decimal places). 1. **Calculate \( \mathrm{S}(2) \):** \[ \mathrm{S}(2) = 0.05(2)^3 + 0.4(2)^2 + 2(2) + 9 = 0.05(8) + 0.4(4) + 4 + 9 = 0.4 + 1.6 + 4 + 9 = 15.00 \text{ million dollars} \] 2. **Calculate \( \mathrm{S}^{\prime}(2) \):** \[ \mathrm{S}^{\prime}(2) = 0.15(2)^2 + 0.8(2) + 2 = 0.15(4) + 1.6 + 2 = 0.6 + 1.6 + 2 = 4.20 \text{ million dollars per month} \] **Answer:** \[ \mathrm{S}(2) = 15.00 \text{ million dollars} \] \[ \mathrm{S}^{\prime}(2) = 4.20 \text{ million dollars per month} \] ### (C) Interpret \( \mathrm{S}(10) = 119.00 \) and \( \mathrm{S}^{\prime}(10) = 25.00 \). 1. **Interpretation of \( \mathrm{S}(10) = 119.00 \):** - This means that **10 months from now**, the total sales are projected to be **\$119 million**. 2. **Interpretation of \( \mathrm{S}^{\prime}(10) = 25.00 \):** - This indicates that **at the 10th month**, the sales are **increasing at a rate of \$25 million per month**. **Answer:** - **(A)** \( \mathrm{S}^{\prime}(t) = 0.15\, t^{2} + 0.8\, t + 2 \) - **(B)** \( \mathrm{S}(2) = 15.00 \) million dollars and \( \mathrm{S}^{\prime}(2) = 4.20 \) million dollars per month - **(C)** At 10 months, total sales are projected to be \$119.00 million, and the sales are increasing at a rate of \$25.00 million per month.