To find out when Valeria and Autumn will be on the same page of the book, we can represent their reading progress with equations based on the information given.
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Valeria's Reading Progress:
- Initial page = 20
- Reading speed = 20 pages per day
- After \( t \) days, her page number \( V(t) \) can be expressed as: \[ V(t) = 20 + 20t \]
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Autumn's Reading Progress:
- Initial page = 40
- Reading speed = 15 pages per day
- After \( t \) days, her page number \( A(t) \) can be expressed as: \[ A(t) = 40 + 15t \]
Next, we need to find when both are on the same page, so we set \( V(t) \) equal to \( A(t) \):
\[ 20 + 20t = 40 + 15t \]
Now, let's solve for \( t \):
\[ 20t - 15t = 40 - 20 \]
\[ 5t = 20 \]
\[ t = 4 \]
Thus, Valeria and Autumn will be on the same page after 4 days.
Verification:
- After 4 days:
- Valeria's page: \( V(4) = 20 + 20(4) = 20 + 80 = 100 \)
- Autumn's page: \( A(4) = 40 + 15(4) = 40 + 60 = 100 \)
Both Valeria and Autumn are on page 100 after 4 days, confirming our solution.
Graphing:
To graph \( V(t) \) and \( A(t) \):
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Valeria's Function: \( V(t) = 20 + 20t \)
- At \( t = 0 \), \( V(0) = 20 \)
- At \( t = 4 \), \( V(4) = 100 \)
This is a straight line with a slope of 20, starting at (0, 20).
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Autumn's Function: \( A(t) = 40 + 15t \)
- At \( t = 0 \), \( A(0) = 40 \)
- At \( t = 4 \), \( A(4) = 100 \)
This is also a straight line but with a lower slope of 15, starting at (0, 40).
The two lines intersect at \( t = 4 \), confirming they will be on the same page of the book after 4 days.
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