To find when \( f(x) = g(x) \), we need to set the two functions equal to each other:
\[ f(x) = 60 + 50x \] \[ g(x) = 2^x \]
Setting them equal gives us:
\[ 60 + 50x = 2^x \]
To find the solution, let's solve this equation graphically. First, we will graph both functions separately and find where they intersect.
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Graphing \( f(x) = 60 + 50x \): This is a linear function with a y-intercept of 60 and a slope of 50. It will increase continuously as \( x \) increases.
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Graphing \( g(x) = 2^x \): This is an exponential function that starts at (0, 1) and increases rapidly for positive values of \( x \).
To find where these two functions intersect, we can evaluate both functions for various integer values of \( x \):
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For \( x = 0 \):
\( f(0) = 60 + 50 \cdot 0 = 60 \)
\( g(0) = 2^0 = 1 \) -
For \( x = 1 \):
\( f(1) = 60 + 50 \cdot 1 = 110 \)
\( g(1) = 2^1 = 2 \) -
For \( x = 2 \):
\( f(2) = 60 + 50 \cdot 2 = 160 \)
\( g(2) = 2^2 = 4 \) -
For \( x = 3 \):
\( f(3) = 60 + 50 \cdot 3 = 210 \)
\( g(3) = 2^3 = 8 \) -
For \( x = 4 \):
\( f(4) = 60 + 50 \cdot 4 = 260 \)
\( g(4) = 2^4 = 16 \) -
For \( x = 5 \):
\( f(5) = 60 + 50 \cdot 5 = 310 \)
\( g(5) = 2^5 = 32 \) -
For \( x = 6 \):
\( f(6) = 60 + 50 \cdot 6 = 360 \)
\( g(6) = 2^6 = 64 \) -
For \( x = 7 \):
\( f(7) = 60 + 50 \cdot 7 = 410 \)
\( g(7) = 2^7 = 128 \) -
For \( x = 8 \):
\( f(8) = 60 + 50 \cdot 8 = 460 \)
\( g(8) = 2^8 = 256 \) -
For \( x = 9 \):
\( f(9) = 60 + 50 \cdot 9 = 510 \)
\( g(9) = 2^9 = 512 \) -
For \( x = 10 \):
\( f(10) = 60 + 50 \cdot 10 = 560 \)
\( g(10) = 2^{10} = 1024 \)
From our calculations, we can see:
- At \( x = 9 \), \( f(9) = 510 \) and \( g(9) = 512 \).
- At \( x = 8 \), \( f(8) = 460 \) and \( g(8) = 256 \).
Therefore, the two values are very close at \( x = 9 \).
Thus, the answer is: A. 9 weeks.