Question

A company's sales for a new product during the first 6 days can be modeled by the function shown in the table, where `S(x)` represents the number of sales (in hundreds) at the end of each day. Find the average rate of change in the sales from day 3 to day 5.

Day S(x)
1 2
2 4
3 8
4 16
5 32
6 64
7 128
Responses

$1,200 per day
$1,200 per day

$2,400 per day
$2,400 per day

$24 per day
$24 per day

$12 per day

Answer 1

To calculate the average rate of change in sales from day 3 to day 5, we use the formula for average rate of change:

\[
\text{Average Rate of Change} = \frac{S(b) - S(a)}{b - a}
\]

where \( S(x) \) is the sales at the end of day \( x \), \( a \) is the initial day and \( b \) is the final day.

In this case, we have:

- \( a = 3 \)
- \( b = 5 \)

From the table, we need to find \( S(3) \) and \( S(5) \):

- \( S(3) = 8 \) (which means 800 sales)
- \( S(5) = 32 \) (which means 3200 sales)

Now we can calculate the average rate of change:

\[
\text{Average Rate of Change} = \frac{S(5) - S(3)}{5 - 3} = \frac{32 - 8}{5 - 3} = \frac{24}{2} = 12
\]

Since \( S(x) \) represents sales in hundreds, the average rate of change is:

\[
12 \times 100 = 1200 \quad \text{sales per day}
\]

Thus, the average rate of change in the sales from day 3 to day 5 is:

\(\$1,200\) per day.

So the correct answer is:

**$1,200 per day**.