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?

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

GPT-4o mini · Answer

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:

$60 + 50 x = 2^{x}$

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.

Graphing the functions:
- For $f (x) = 60 + 50 x$ : 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.
Finding the intersection numerically:

You can use graphing software, a graphing calculator, or numerical methods to solve $60 + 50 x = 2^{x}$ . 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