A mosquito control truck is spraying pesticides to control the population of mosquitoes in an area. The function [tex] p(t) = 15,000 \left( \frac{1}{7} \right)^t [/tex] represents the population of mosquitoes, [tex] p(t) [/tex], after each application of pesticides, [tex] t [/tex]. Does the function represent growth or decay?

A. The function represents exponential growth because the base equals 15,000.
B. The function represents exponential decay because the base equals 15,000.
C. The function represents exponential growth because the base equals [tex] \frac{1}{7} [/tex].
D. The function represents exponential decay because the base equals [tex] \frac{1}{7} [/tex].



Answer :

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]