Question

At the beginning of the year, Noah had $90 in savings and saved an additional $10 each week thereafter. Jace started the year with $50 and saved $20 every week. Let NN represent the amount of money Noah has saved tt weeks after the beginning of the year and let JJ represent the amount of money Jace has saved tt weeks after the beginning of the year. Graph each function and determine the number of weeks after the beginning of the year until Noah and Jace have the same amount of money saved.

Answer 1

Noah and Jace will have the same amount of money saved 4 weeks after the beginning of the year. This is the point of intersection on the graph of the two functions.

Explanation:

To graph the functions that represent the amount of money Noah and Jace have saved, we can set up their savings functions as follows:

Noah's savings function (NN):

NN(tt) = $90 (initial savings) + $10 (weekly savings) * tt

Jace's savings function (JJ):

JJ(tt) = $50 (initial savings) + $20 (weekly savings) * tt

Now, let's graph these functions:

Noah's Savings Function (NN):

Initial savings (y-intercept): $90

Weekly savings rate (slope): $10

Jace's Savings Function (JJ):

Initial savings (y-intercept): $50

Weekly savings rate (slope): $20

To find the number of weeks after the beginning of the year until Noah and Jace have the same amount of money saved, we can set the two functions equal to each other and solve for "tt":

NN(tt) = JJ(tt)

$90 + $10 * tt = $50 + $20 * tt

Now, solve for "tt":

$90 - $50 = $20 * tt - $10 * tt

$40 = $10 * tt

tt = $40 / $10

tt = 4