To find out when Kelly and Eddie will have saved the same amount of money, we need to set the two functions equal to each other:
\[ f(x) = g(x) \]
Using the functions given:
\[ 60 + 50x = 2^x \]
Next, we'll solve this equation for \( x \). Because we have a linear equation on one side and an exponential equation on the other, we may need to use numerical methods or graphing to find the intersection point.
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Graphing the functions:
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For \( f(x) = 60 + 50x \): This is a linear function that starts at \( 60 \) when \( x = 0 \) and increases by \( 50 \) for each additional week.
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For \( g(x) = 2^x \): This is an exponential function that starts at \( 1 \) when \( x = 0 \) and grows rapidly as \( x \) increases.
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Finding the intersection numerically:
You can use graphing software, a graphing calculator, or numerical methods to solve \( 60 + 50x = 2^x \). However, we'll also try a few values by hand:
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For \( x = 1 \):
\( f(1) = 60 + 50(1) = 110 \)
\( g(1) = 2^1 = 2 \) -
For \( x = 2 \):
\( f(2) = 60 + 50(2) = 160 \)
\( g(2) = 2^2 = 4 \) -
For \( x = 5 \):
\( f(5) = 60 + 50(5) = 310 \)
\( g(5) = 2^5 = 32 \) -
For \( x = 8 \):
\( f(8) = 60 + 50(8) = 460 \)
\( g(8) = 2^8 = 256 \) -
For \( x = 9 \):
\( f(9) = 60 + 50(9) = 510 \)
\( g(9) = 2^9 = 512 \) -
For \( x = 10 \):
\( f(10) = 60 + 50(10) = 560 \)
\( g(10) = 2^{10} = 1024 \)
From our calculations, we see that:
- At \( x = 8 \), \( f(8) = 460 \) and \( g(8) = 256 \)
- At \( x = 9 \), \( f(9) = 510 \) and \( g(9) = 512 \)
- At \( x = 10 \), \( f(10) = 560 \) and \( g(10) = 1024 \)
The value of \( x \) where they have saved approximately the same amount of money falls between \( 8 \) and \( 9 \). Rounding to the nearest integer, they will have saved the same amount of money after:
Answer: 9 weeks
3 answers