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Determine whether the following equation represents exponential growth or exponential decay.

[tex]\[ y = \left(\frac{11}{30}\right) \cdot \left(\frac{22}{19}\right)^{-x} \][/tex]

A. Exponential decay

B. Exponential growth


Sagot :

To determine whether the given equation represents exponential growth or exponential decay, let's take a detailed step-by-step approach.

The given equation is:
[tex]\[ y = \left(\frac{11}{30}\right) \cdot \left(\frac{22}{19}\right)^{-x} \][/tex]

1. Identify the base of the exponent:
The base of the exponent in the given equation is [tex]\(\frac{22}{19}\)[/tex]. However, the equation has a negative exponent, [tex]\(-x\)[/tex].

2. Understand the effect of the negative exponent:
A negative exponent means that we take the reciprocal of the base. Thus, the term [tex]\(\left(\frac{22}{19}\right)^{-x}\)[/tex] can be rewritten as:
[tex]\[ \left(\frac{22}{19}\right)^{-x} = \left(\frac{19}{22}\right)^{x} \][/tex]

3. Determine the value of the new base:
After taking the reciprocal, the new base is [tex]\(\frac{19}{22}\)[/tex].

4. Compare the new base to 1:
- If the base is greater than 1, it indicates exponential growth.
- If the base is less than 1, it indicates exponential decay.

Since [tex]\(\frac{19}{22} < 1\)[/tex], we see that the base is less than 1.

Hence, the equation:
[tex]\[ y = \left(\frac{11}{30}\right) \cdot \left(\frac{19}{22}\right)^{x} \][/tex]

With a base ([tex]\(\frac{19}{22}\)[/tex]) less than 1, this equation represents exponential decay.

Conclusion:
The given equation represents exponential decay.