To find out when Valeria and Autumn will be on the same page of the book, we can start by defining the functions for the pages they are each on after \( t \) days.
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Valeria's page function, \( V(t) \):
- She starts on page 20 and reads 20 pages per day.
- The function can be represented as: \[ V(t) = 20 + 20t \]
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Autumn's page function, \( A(t) \):
- She starts on page 40 and reads 15 pages per day.
- The function can be represented as: \[ A(t) = 40 + 15t \]
Next, we set these two functions equal to find out when they will be on the same page: \[ 20 + 20t = 40 + 15t \]
Now, we'll solve for \( t \):
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Subtract \( 15t \) from both sides: \[ 20 + 20t - 15t = 40 \] \[ 20 + 5t = 40 \]
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Subtract 20 from both sides: \[ 5t = 20 \]
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Divide both sides by 5: \[ t = 4 \]
Thus, Valeria and Autumn will be on the same page after 4 days.
Verification
We can check what page they will be on after 4 days:
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For Valeria: \[ V(4) = 20 + 20(4) = 20 + 80 = 100 \]
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For Autumn: \[ A(4) = 40 + 15(4) = 40 + 60 = 100 \]
Both will be on page 100 after 4 days.
Conclusion
Valeria and Autumn will be on the same page of the book after \( t = 4 \) days.