To determine whether the function [tex]\(p(t) = 15,000 \left(\frac{1}{7}\right)^t\)[/tex] represents growth or decay, we need to focus on the base of the exponential expression.
1. Observe the given function:
[tex]\[
p(t) = 15,000 \left(\frac{1}{7}\right)^t
\][/tex]
2. Identify the base of the exponential term:
[tex]\[
\left(\frac{1}{7}\right)^t
\][/tex]
3. Determine whether the base indicates growth or decay:
- Exponential functions generally have the form [tex]\(a \cdot b^t\)[/tex].
- If the base [tex]\(b\)[/tex] (in this case, [tex]\(\frac{1}{7}\)[/tex]) is greater than 1, the function represents exponential growth.
- If the base [tex]\(b\)[/tex] is between 0 and 1, the function represents exponential decay.
4. Evaluate the base [tex]\(\frac{1}{7}\)[/tex]:
[tex]\[
\frac{1}{7} \approx 0.142857
\][/tex]
Since [tex]\(\frac{1}{7}\)[/tex] is less than 1 but greater than 0, it falls into the range that signifies decay.
5. Conclusion:
The function represents exponential decay because the base equals [tex]\(\frac{1}{7}\)[/tex].
Therefore, the correct conclusion is:
[tex]\[
\boxed{\text{The function represents exponential decay because the base equals } \frac{1}{7}.}
\][/tex]
