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}
s(t)=\frac{16000t}{(t^{2}+100)^{2}}
b) find s(5) and s(5).
the value of s(5) rounded to the nearest hundredth is 16.00.
the value of s(5) rounded to the nearest hundredth is 5.12.
what do the values for s(5) and s(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.
c) use the results from part (b) to estimate the total sales after 6 months.
after 6 months, the total sales will be approximately how many dvds?
(round to the nearest whole number.)

Explanation:

Step1: Interpret $S(5)$ and $S^{\prime}(5)$

$S(5)$ represents total sales at $t = 5$ months and $S^{\prime}(5)$ represents rate of change of sales at $t=5$ months. Given $S(5)=16.00$ (in thousands) and $S^{\prime}(5) = 5.12$ (in thousands per month).

Step2: Estimate sales at $t = 6$ months

We use the linear - approximation formula $S(6)\approx S(5)+S^{\prime}(5)\times(6 - 5)$. Substitute $S(5)=16.00$ (in thousands) and $S^{\prime}(5)=5.12$ (in thousands). So $S(6)\approx16.00 + 5.12\times1=21.12$ (in thousands).

Step3: Convert to whole number

Since $S$ is in thousands, multiply by 1000. $21.12\times1000 = 21120$.

Answer:

21120