Question

the total sales of s (in thousands of dvds) of a certain movie are given by the following formula where t is the number of months since the release of the dvd. use the formula to answer the questions.
s(t)=\frac{80t^{2}}{t^{2}+100}
a) find (s^{prime}(t)).
s^{prime}(t)=\frac{16000t}{(t^{2}+100)^{2}}
b) find (s(5)) and (s^{prime}(5)).
the value of (s(5)) rounded to the nearest hundredth is 16.00.
the value of (s^{prime}(5)) rounded to the nearest hundredth is 5.12.
what do the values for (s(5)) and (s^{prime}(5)) indicate?
a. after 5 months, the total sales are 51,200 dvds and the sales are increasing at the rate of 1600 dvds per month.
b. after 5 months, the total sales are 16,000 dvds and the sales are increasing at the rate of 5.12 dvds per month.
c. after 5 months, the total sales are 16,000 dvds and the sales are increasing at the rate of 5120 dvds per month.
d. after 5 months, the total sales are 5120 dvds and the sales are increasing at the rate of 16 dvds per month.

Explanation:

Step1: Interpret the function S(t)

$S(t)$ gives total sales in thousands of DVDs and $S'(t)$ is the rate of change of sales.

Step2: Analyze S(5)

Since $S(t)$ is in thousands of DVDs, when $S(5)=16.00$, the actual number of DVDs sold is $16.00\times1000 = 16000$.

Step3: Analyze S'(5)

$S'(5) = 5.12$ means the rate of change of sales at $t = 5$ months is 5.12 thousand DVDs per month.

Answer:

B. After 5 months, the total sales are 16,000 DVD's and the sales are increasing at the rate of 5.12 DVD's per month.