Write an exponential equation in the form y = a(b)^x that can model Riley's target mileage per week, y, x weeks after she starts training.
Understand the Problem
The question is asking us to write an exponential equation in the form y = a(b)^x, where y represents the target mileage and x represents the number of weeks after training starts. The initial mileage is 12 miles, and it increases by 10% each week.
Answer
The equation modeling Riley's target mileage is $y = 12(1.10)^x$.
Answer for screen readers
The exponential equation that models Riley's target mileage is given by: $$ y = 12(1.10)^x $$
Steps to Solve
- Identify the Initial Value ($a$)
The initial mileage is 12 miles. This represents the value of $a$ in the equation.
So, we have: $$ a = 12 $$
- Determine the Growth Factor ($b$)
Riley's mileage increases by 10% each week. To express this as a growth factor, we convert the percentage to a decimal and add 1.
Thus: $$ b = 1 + 0.10 = 1.10 $$
- Write the Exponential Equation
Now we can substitute the values of $a$ and $b$ into the exponential equation format $y = a(b)^x$.
The equation becomes: $$ y = 12(1.10)^x $$
The exponential equation that models Riley's target mileage is given by: $$ y = 12(1.10)^x $$
More Information
In this context, the exponential equation signifies that Riley's mileage increases by 10% each week, starting from an initial run of 12 miles. This model can help track her progress and predict future mileage over the training weeks.
Tips
- Miscalculating the growth factor: Remember that a 10% increase means multiplying by 1.10, not just adding 10.
- Forgetting to write the equation in the proper exponential form.
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