Question

Sophie and Max are saving money for a road trip. The total amount of money, in dollars, that Sophie will save over x weeks is modeled by the function f(x) = 80 + 30x. The total amount of money, in dollars, that Max will save over x weeks is modeled by the function g(x) = 3^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?

Answer 1

To determine when Sophie and Max will have saved the same amount of money, we need to find the value of \( x \) for which \( f(x) = g(x) \).

The functions are given as:
- Sophie: \( f(x) = 80 + 30x \)
- Max: \( g(x) = 3^x \)

We want to set these two equations equal to each other and solve for \( x \):

\[
80 + 30x = 3^x
\]

Since this equation is not easily solvable algebraically, we'll find the solution by graphing both functions and looking for their intersection point.

### Step 1: Create a graph of the functions

1. **Sophie’s savings (f(x))**:
- This is a linear function starting at \( 80 \) when \( x = 0 \), with a slope of \( 30 \). It will rise steadily.

2. **Max’s savings (g(x))**:
- This is an exponential function. Starting at \( 3^0 = 1 \), it increases more rapidly as \( x \) increases.

### Step 2: Graphing

You can use a graphing tool or software like Desmos, GeoGebra, or even a graphing calculator to plot both functions.

In the coordinate plane:
- For \( f(x) = 80 + 30x \), plot points such as:
- \( f(0) = 80 \)
- \( f(1) = 110 \)
- \( f(2) = 140 \)
- \( f(3) = 170 \)
- Continue calculating and plotting for a few more points.

- For \( g(x) = 3^x \), plot points such as:
- \( g(0) = 1 \)
- \( g(1) = 3 \)
- \( g(2) = 9 \)
- \( g(3) = 27 \)
- \( g(4) = 81 \)
- \( g(5) = 243 \)

### Step 3: Observing the Intersection

By looking at the graph, we should find the approximate value of \( x \) where the two graphs intersect.

### Step 4: Solving with a numerical approach

Alternatively, you can solve this numerically by trying different values for \( x \):

- \( x = 5 \):
- \( f(5) = 80 + 30(5) = 80 + 150 = 230 \)
- \( g(5) = 3^5 = 243 \)

- \( x = 4 \):
- \( f(4) = 80 + 30(4) = 80 + 120 = 200 \)
- \( g(4) = 3^4 = 81 \)

From these calculations, we see that \( f(5) = 230 \) and \( g(5) = 243 \) are quite close together, so the intersection must be between \( x = 4 \) and \( x = 5 \).

We can try \( x = 4.5 \):
- \( f(4.5) = 80 + 30(4.5) = 80 + 135 = 215 \)
- \( g(4.5) = 3^{4.5} \approx 140.3 \)

### Final Approximation

Continuing to narrow this down by interpolating between the values, we'd find the two functions intersect at approximately \( x \approx 4.8 \).

### Conclusion

Rounding to the nearest integer, they will have saved the same amount of money after **5 weeks**.