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.

Explanation:

Step1: Differentiate $S(t)$ to find $S'(t)$

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

Step2: Calculate $S(5)$

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

Step3: Calculate $S'(5)$

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

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

$S(8) = 107.12$ means that 8 months from now, the total sales of the company will be 107.12 million dollars.
$S'(8)=17.92$ means that 8 months from now, the rate of change of the company's total sales is 17.92 million dollars per month.

Answer:

(A) $S'(t)=0.03t^{2}+t + 8$
(B) $S(5)=59.75$, $S'(5)=13.75$
(C) $S(8) = 107.12$ means the total sales 8 months from now is 107.12 million dollars; $S'(8)=17.92$ means the rate of change of total sales 8 months from now is 17.92 million dollars per month.