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Question Given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. \[ y=33(0.492)^{x} \] Answer Attempt 1 out of 2 Growth \% increase Submit Answer
Ask by Young Grant. in the United States
Feb 25,2025
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Tutor-Verified Answer
Answer
The function represents exponential decay with a 50.8% decrease per unit increase in x.
Solution
To determine whether the function represents growth or decay, look at the base of the exponential term:
y = 33(0.492)^x
Since the base 0.492 is less than 1, the function is decreasing (decay).
Next, to find the percentage rate of decrease, note that when a quantity is multiplied by a number less than 1, the decrease is given by 1 minus that number:
Percentage decrease = (1 - 0.492) × 100% = 0.508 × 100% = 50.8%
Thus, the function represents exponential decay, and the rate of decrease is approximately 50.8% per unit increase in x.
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Simplify this solution Beyond the Answer
This exponential function represents decay since the base (0.492) is less than 1. In this scenario, we are looking at a situation where the quantity decreases over time. To determine the percentage rate of decrease, you can use the formula \( \text{percentage rate} = (1 - \text{base}) \times 100 \). Here, it would be \( (1 - 0.492) \times 100 = 50.8\% \). So, this function reflects a decay rate of approximately 50.8%.
