Let's analyze the given problem step by step:
### Given:
The total sales function [tex]\( S(t) \)[/tex] (in thousands of DVDs) is described by:
[tex]\[ S(t) = \frac{40 t^2}{t^2 + 150} \][/tex]
### Part (a): Finding the derivative [tex]\( S'(t) \)[/tex]
We are given the derivative:
[tex]\[ S'(t) = \frac{-80 t^3}{(t^2 + 150)^2} + \frac{80 t}{t^2 + 150} \][/tex]
### Part (b): Finding [tex]\( S(10) \)[/tex] and [tex]\( S'(10) \)[/tex]
Substitute [tex]\( t = 10 \)[/tex] into the original function [tex]\( S(t) \)[/tex]:
[tex]\[
S(10) = \frac{40 \cdot 10^2}{10^2 + 150} = \frac{40 \cdot 100}{100 + 150} = \frac{4000}{250} = 16.0 \, \text{(in thousands of DVDs)}
\][/tex]
Then, substitute [tex]\( t = 10 \)[/tex] into the derivative [tex]\( S'(t) \)[/tex]:
[tex]\[
S'(10) = \frac{-80 \cdot 10^3}{(10^2 + 150)^2} + \frac{80 \cdot 10}{10^2 + 150} = \frac{-80 \cdot 1000}{(100 + 150)^2} + \frac{800}{250} = \frac{-80000}{62500} + \frac{800}{250}
\][/tex]
Calculating the above expressions step-by-step:
[tex]\[
\frac{-80000}{62500} = -1.28
\][/tex]
[tex]\[
\frac{800}{250} = 3.20
\][/tex]
Combining these together:
[tex]\[
S'(10) = -1.28 + 3.20 = 1.92 \, \text{(in thousands of DVDs per month)}
\][/tex]
### Interpretation of the values:
- [tex]\( S(10) = 16.0 \)[/tex] thousand DVDs means that after 10 months, the total sales are:
[tex]\[ 16.0 \times 1000 = 16000 \][/tex] DVDs
- [tex]\( S'(10) = 1.92 \)[/tex] thousand DVDs per month means the rate at which sales are increasing after 10 months is:
[tex]\[ 1.92 \times 1000 = 1920 \][/tex] DVDs per month
### Choosing the correct interpretation:
From the given choices:
- A: Incorrect, the total sales after 10 months are not 19,200 DVDs.
- B: Incorrect, the total sales after 10 months are not 1,920 DVDs, and the rate of increase is not 16 DVDs per month.
- C: Incorrect, the total sales are correct at 16,000 DVDs, but the rate of increase is not 1.92 DVDs per month.
- D: Correct, after 10 months, the total sales are 16,000 DVDs, and the sales are increasing at the rate of 1920 DVDs per month.
Thus, the correct interpretation of the values is:
[tex]\[ \text{D. After 10 months, the total sales are 16,000 DVDs and the sales are increasing at the rate of 1920 DVDs per month.} \][/tex]
