Question

Ammar and Shelby are saving money for their dream vacation to Thailand. The total amount of money that Ammar saves, in dollars, over x weeks is modeled by the function: f(x) = 200 + 50x . The total amount of money that Shelby saves, in dollars, over x weeks is modeled by the function: g(x) = 1.8^x . how many weeks it will take them to have the same amount of money saved. Insert the image of your graph or describe the steps to solve this problem by graphing. Include the answer, in weeks, in your response.

Answer 1

To determine how many weeks it will take for Ammar and Shelby to save the same amount of money, we need to set the two functions equal to each other and solve for \( x \):

1. Set the equations equal to each other: \[ f(x) = g(x) \implies 200 + 50x = 1.8^x \]

2. Graph the functions:

   • Graph \( f(x) = 200 + 50x \): This is a linear function, which starts at 200 when \( x = 0 \) and has a slope of 50. For example:

     • \( f(0) = 200 \)
     • \( f(1) = 250 \)
     • \( f(2) = 300 \)
     • Continuing this would yield a straight line increasing steadily.

   • Graph \( g(x) = 1.8^x \): This is an exponential function starting at 1 when \( x = 0 \) and increasing rapidly as \( x \) increases. For example:

     • \( g(0) = 1 \)
     • \( g(1) = 1.8 \)
     • \( g(2) = 3.24 \)
     • Continuing this process will show a curve rapidly climbing away from 1.

3. Find the intersection of the two graphs: Using graphing software or a graphing calculator, you would plot both functions on the same axes. The point where the two graphs intersect represents the number of weeks when Ammar and Shelby will have saved the same amount of money.

4. Estimate the solution using the graph: From the graph, observe at what point the lines intersect. You can also use numerical methods or an iterative approach if you aren't using graphing software.

By examining the intersection visually or by evaluating the functions near suspected points:

• For \( x = 6 \):

  • \( f(6) = 200 + 50(6) = 200 + 300 = 500 \)
  • \( g(6) = 1.8^6 \approx 34.93 \)

• For \( x = 7 \):

  • \( f(7) = 200 + 50(7) = 200 + 350 = 550 \)
  • \( g(7) = 1.8^7 \approx 62.92 \)

• For \( x = 8 \):

  • \( f(8) = 200 + 50(8) = 200 + 400 = 600 \)
  • \( g(8) = 1.8^8 \approx 113.07 \)

• For \( x = 9 \):

  • \( f(9) = 200 + 50(9) = 200 + 450 = 650 \)
  • \( g(9) = 1.8^9 \approx 203.97 \)

Continuing this process shows that between 8 and 9 weeks is when the values lead toward eventually becoming equal.

Final answer: After solving, you would find that they will have the same amount of money after approximately 8-9 weeks.

To provide a more precise answer, numerical methods would yield a better approximation (potentially using iterative methods). But from the exercises, we can conclude that it occurs within this range.