Write an exponential equation in the form y = a(b)^x that can model the value of Nora's motorcycle, y, x years after purchase.
Understand the Problem
The question is asking to formulate an exponential equation to model the decreasing value of Nora's motorcycle over time, using the given initial value and the value after one year. We will identify parameters a and b to create the equation in the required form.
Answer
The equation for the value of Nora's motorcycle is $y = 12000(0.88)^x$.
Answer for screen readers
The exponential equation that models the value of Nora's motorcycle is: $$ y = 12000(0.88)^x $$
Steps to Solve
- Identify initial value (a)
The initial value of the motorcycle is given as $12,000. Thus, we have: $$ a = 12000 $$
- Determine the value after one year (y)
The value of the motorcycle after one year is $10,560. This value will help us find the decay factor (b). So: $$ y = 10560 $$
- Set up the equation for one year
Using the exponential model ( y = a(b)^x ) for ( x = 1 ): $$ 10560 = 12000(b)^1 $$
- Solve for b
Now we need to solve for (b) by isolating it: First, divide both sides by (12000): $$ b = \frac{10560}{12000} $$
Now, perform the division:
$$ b = 0.88 $$
- Write the final equation
Now that we have (a) and (b), we can write the exponential equation: $$ y = 12000(0.88)^x $$
The exponential equation that models the value of Nora's motorcycle is: $$ y = 12000(0.88)^x $$
More Information
This equation indicates that each year, the value of the motorcycle decreases by 12%. An (b) value less than 1 signifies a decay in value.
Tips
- Forgetting to convert the value after one year into a fraction relative to the initial price can lead to an incorrect decay factor (b).
- Starting with (x = 0) instead of (x = 1) when calculating the value after the first year can cause confusion.
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