Determine whether the following equation represents an exponential growth or exponential decay.
[tex]y=288.5 \cdot(0.84)^x[/tex]
A. exponential growth
B. exponential decay
Answer (2)
e x p o n e n t ia l d ec a y
Explanation
Identifying the Equation Type The given equation is y = 288.5 ⋅ ( 0.84 ) x . We need to determine whether this equation represents exponential growth or exponential decay.
Understanding Exponential Growth and Decay An exponential function is of the form y = a b x , where a is the initial value and b is the base. If 1"> b > 1 , the function represents exponential growth, and if 0 < b < 1 , the function represents exponential decay.
Analyzing the Base In our equation, y = 288.5 ⋅ ( 0.84 ) x , we can see that the base b is 0.84 . Since 0 < 0.84 < 1 , the equation represents exponential decay.
Conclusion Therefore, the equation y = 288.5 ⋅ ( 0.84 ) x represents exponential decay.
Examples
Exponential decay is a mathematical concept that has many real-world applications. For example, the depreciation of a car's value over time can be modeled using exponential decay. If a car is bought for 25 , 000 an dd e p rec ia t es a t a r a t eo f 15 V = 25000 \cdot (0.85)^t$, where V is the value of the car after t years. Understanding exponential decay helps in making informed decisions about investments, loans, and asset management.
The equation y = 288.5 ⋅ ( 0.84 ) x represents exponential decay because its base, 0.84, is between 0 and 1. This indicates that the function will decrease in value as x increases. Therefore, the correct answer is B. exponential decay.
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