To model the pounds of candy on sale after October as given in the data table, we note the pattern that the amount of candy is decreasing each week. We need to find an exponential equation of the form [tex]\( A = P \cdot (r^x) \)[/tex], where:
- [tex]\( A \)[/tex] is the amount of candy after [tex]\( x \)[/tex] weeks,
- [tex]\( P \)[/tex] is the initial amount of candy,
- [tex]\( r \)[/tex] is the common ratio by which the candy amount decreases weekly,
- [tex]\( x \)[/tex] is the number of weeks.
Here's the step-by-step process to find the equation:
1. Identify the initial amount [tex]\( P \)[/tex]:
The given data shows the amount of candy at week 0 (initial amount) is [tex]\( 990 \)[/tex] pounds. So, [tex]\( P = 990 \)[/tex].
2. Calculate the common ratio [tex]\( r \)[/tex]:
The common ratio can be calculated using the amounts given for consecutive weeks. For instance, the amount of candy decreases from [tex]\( 990 \)[/tex] pounds in week 0 to [tex]\( 742.5 \)[/tex] pounds in week 1. The common ratio [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{\text{Amount in Week 1}}{\text{Amount in Week 0}} = \frac{742.5}{990} = 0.75
\][/tex]
3. Formulate the exponential decay equation:
Using the initial amount [tex]\( P = 990 \)[/tex] and the common ratio [tex]\( r = 0.75 \)[/tex], the equation modeling the data is:
[tex]\[
A = 990 \cdot (0.75^x)
\][/tex]
Thus, the equation that models the pounds of candy on sale after [tex]\( x \)[/tex] weeks is:
[tex]\[
A = 990 \cdot (0.75^x)
\][/tex]
This equation accurately reflects the given data and describes how the amount of candy decreases exponentially over time.
