Question

the total sales of a company (in millions of dollars) t months from now are given by s(t)=0.01t³ + 0.5t² + 8t + 6. (a) find s′(t). (b) find s(5) and s′(5) (to two decimal places). (c) interpret s(8)=107.12 and s′(8)=17.92. a. the total sales 8 months from now will be $107.12 million dollars. b. in approximately 107 months the sales will be increasing by $8 million per month. c. in 8 months the sales will be increasing at a rate of $107.12 million per month. d. in approximately 107 months the sales will be $8 million dollars. choose the correct interpretation of s′(8)=17.92. a. in 8 months the sales will be increasing at a rate of $17.92 million per month. b. the total sales 8 months from now will be $17.92 million dollars. c. in approximately 18 months the sales will be increasing by $8 million per month. d. in approximately 18 months the sales will be $8 million dollars.

Explanation:

Step1: Differentiate S(t)

Using the power - rule $\frac{d}{dt}(t^n)=nt^{n - 1}$, we have $S(t)=0.01t^{3}+0.5t^{2}+8t + 6$. Then $S'(t)=0.01\times3t^{2}+0.5\times2t+8=0.03t^{2}+t + 8$.

Step2: Calculate S(5)

Substitute $t = 5$ into $S(t)$: $S(5)=0.01\times5^{3}+0.5\times5^{2}+8\times5 + 6=0.01\times125+0.5\times25 + 40+6=1.25 + 12.5+40+6=59.75$.

Step3: Calculate S'(5)

Substitute $t = 5$ into $S'(t)$: $S'(5)=0.03\times5^{2}+5 + 8=0.03\times25+5 + 8=0.75+5 + 8=13.75$.

Step4: Interpret S(8) and S'(8)

$S(8)$ represents the total sales (in millions of dollars) 8 months from now. So, when $S(8)=107.12$, it means the total sales 8 months from now will be $\$107.12$ million. $S'(t)$ represents the rate of change of sales (in millions of dollars per month). So, when $S'(8)=17.92$, it means in 8 months the sales will be increasing at a rate of $\$17.92$ million per month.

Answer:

(A) $S'(t)=0.03t^{2}+t + 8$
(B) $S(5)=59.75$, $S'(5)=13.75$
(C) For $S(8)=107.12$, the total sales 8 months from now will be $\$107.12$ million; for $S'(8)=17.92$, in 8 months the sales will be increasing at a rate of $\$17.92$ million per month.
For the multiple - choice part:
Interpretation of $S(8)=107.12$: A. The total sales 8 months from now will be $\$107.12$ million dollars.
Interpretation of $S'(8)=17.92$: A. In 8 months the sales will be increasing at a rate of $\$17.92$ million per month.