The total sales of a company (in millions of dollars) \( t \) months from now are giv (A) Find \( S^{\prime}(t) \). (B) Find \( S(2) \) and \( S^{\prime}(2) \) (to two decimal places). (C) Interpret \( S(10)=119.00 \) and \( S^{\prime}(10)=25.00 \). (A) \( S^{\prime}(t)=0.15 t^{2}+0.8 t+2 \) (B) \( S(2)=\square \) \( S^{\prime}(2)=\square \)

To solve this math problem, let’s delve into the calculations! For part (B): To find \( S(2) \), we need the original function \( S(t) \) derived from \( S^{\prime}(t) \), which is the derivative representing the rate of total sales. Assuming \( S(t) \) is an integral of \( S^{\prime}(t) \): First, we integrate: \[ S(t) = \int (0.15 t^2 + 0.8 t + 2) dt = 0.05 t^3 + 0.4 t^2 + 2t + C \] To find \( S(2) \): \[ S(2) = 0.05 (2)^3 + 0.4 (2)^2 + 2(2) + C = 0.05(8) + 0.4(4) + 4 + C = 0.4 + 1.6 + 4 + C = 6 + C \] Assuming \( C = 0 \), we have \( S(2) = 6 \). To find \( S^{\prime}(2) \): \[ S^{\prime}(2) = 0.15(2^2) + 0.8(2) + 2 = 0.15(4) + 1.6 + 2 = 0.6 + 1.6 + 2 = 4.2 \] Thus, \( S(2) = 6.00 \) \( S^{\prime}(2) = 4.20 \) And for part (C): Interpreting \( S(10) = 119.00 \) means that after 10 months, the company is expected to make $119 million in total sales. \( S^{\prime}(10) = 25.00 \) indicates that at the 10-month mark, the sales are increasing by $25 million per month, suggesting a healthy growth trajectory as they approach this milestone! This function highlights how businesses can analyze future sales and growth patterns, helping them strategize effectively.