To solve this problem, we need to represent the number of termites left after the house is sprayed [tex]\( x \)[/tex] times.
Let's begin with the initial number of termites, which is 12,000. Each time the house is sprayed, the number of termites is reduced to one-fourth of the previous number. We need to express this scenario mathematically.
1. Initial number of termites: 12,000.
2. Effect of one spray: After one spray, the number of termites reduces to one-fourth.
[tex]\[ \text{After 1 spray: } 12,000 \times \left(\frac{1}{4}\right) \][/tex]
3. Effect of two sprays: After two sprays, the remaining termites will be one-fourth of the number after the first spray.
[tex]\[ \text{After 2 sprays: } 12,000 \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = 12,000 \times \left(\frac{1}{4}\right)^2 \][/tex]
4. Effect of three sprays: Similarly, after three sprays:
[tex]\[ \text{After 3 sprays: } 12,000 \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) = 12,000 \times \left(\frac{1}{4}\right)^3 \][/tex]
From these calculations, we can generalize the result for [tex]\( x \)[/tex] sprays. The remaining number of termites after [tex]\( x \)[/tex] sprays would be:
[tex]\[ f(x) = 12,000 \times \left(\frac{1}{4}\right)^x \][/tex]
Let's compare this with the given options:
A. [tex]\( f(x) = 12,000 - \left(\frac{1}{4}\right)^x \)[/tex]
- This implies subtraction of a very small quantity and does not align with the problem statement that suggests multiplication.
B. [tex]\( f(x) = 12,000 \left(\frac{1}{4}\right)^x \)[/tex]
- This matches exactly with our derived general formula.
C. [tex]\( f(x) = \frac{1}{4}(12,000)^x \)[/tex]
- This incorrectly raises 12,000 to the power of [tex]\( x \)[/tex], which is not in line with the pattern of repeated division by 4.
D. [tex]\( f(x) = 12,000 + \left(\frac{1}{4}\right)^x \)[/tex]
- This incorrectly adds a small number instead of multiplying, contrary to our derived pattern.
Thus, the correct function that describes how the number of termites decreases after [tex]\( x \)[/tex] sprays is:
[tex]\[ \boxed{f(x) = 12,000 \left(\frac{1}{4}\right)^x} \][/tex]
