Question

Kelly and eddie are each saving money for a motorcycle. the total amount of money, in dollars, that kelly will save over x weeks is modeled by the function f(x)=60+50x. The total amount of money, in dollars, that eddie will save over x weeks is modeled by the function g(x)=2^x. Graph the functions in the same coordinate plane to determine when f(x)=g(x). After how many weeks, rounded to the nearest integer, will they have saved the same amount of money?
1)510 weeks
2)1 week
3)60 weks
4)9 weeks

Answer 1

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.

1. **Graphing the functions**:

- 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.

- For \( g(x) = 2^x \): This is an exponential function that starts at \( 1 \) when \( x = 0 \) and grows rapidly as \( x \) increases.

2. **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:

- **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