Tim explained a function in words and Paul wrote an equation.
Tim: The amount of money in a savings account increases at a rate of $225 per month. After eight months, the bank account has $4,580 in it.
Paul: [tex]y-1,400=56(x+26)[/tex]
Whose function has the smaller [tex]y[/tex]-intercept?
A. Tim's with a [tex]y[/tex]-intercept of $2,700
B. Paul's with a [tex]y[/tex]-intercept of $2,856
C. Paul's with a [tex]y[/tex]-intercept of $2,800
D. Tim's with a [tex]y[/tex]-intercept of $2,780
Answer (2)
Determine Tim's function: y = 225 x + b , and calculate the y-intercept: b = 4580 − 225 ( 8 ) = 2780 .
Determine Paul's function: y = 56 x + 1400 + 56 ( 26 ) , and calculate the y-intercept: y = 56 x + 2856 .
Compare the y-intercepts: 2780 < 2856 .
Tim's function has the smaller y-intercept: Tim’s with a y -intercept of $2 , 780
Explanation
Analyze Tim's Function Let's analyze Tim's function first. We know the account increases at a rate of $225 per month, and after 8 months, the account has 4 , 580. W ec an e x p ress t hi s a s a l in e a re q u a t i o n w h ere y i s t h e am o u n t in t h e a cco u n t an d x$ is the number of months. The slope of the line is the rate of increase, which is 225. S o , w e ha v e y = 225x + b , w h ere b$ is the y-intercept (the initial amount in the account).
Calculate Tim's y-intercept To find the y-intercept for Tim's function, we can plug in the given point (8, 4580) into the equation: 4580 = 225 ( 8 ) + b . Solving for b , we get: 4580 = 1800 + b b = 4580 − 1800 b = 2780 . So, Tim's function has a y-intercept of $2780.
Analyze and calculate Paul's y-intercept Now let's analyze Paul's function: y − 1400 = 56 ( x + 26 ) . To find the y-intercept, we need to rewrite the equation in slope-intercept form ( y = m x + b ). y − 1400 = 56 x + 56 ( 26 ) y − 1400 = 56 x + 1456 y = 56 x + 1456 + 1400 y = 56 x + 2856 . So, Paul's function has a y-intercept of $2856.
Compare y-intercepts and conclude Comparing the two y-intercepts, Tim's function has a y-intercept of $2780, and Paul's function has a y-intercept of $2856. Since 2780 < 2856 , Tim's function has the smaller y-intercept.
Examples
Understanding y-intercepts is crucial in various real-life scenarios, such as determining initial costs or values. For instance, in business, the y-intercept of a cost function represents the fixed costs before any units are produced. Similarly, in physics, it can represent the initial position of an object. In this problem, we found the initial amount in Tim's savings account and compared it to a similar value in Paul's equation, illustrating how linear functions model real-world situations.
After calculating both Tim's and Paul's functions, Tim's function has a y-intercept of $2,780, while Paul's function has a y-intercept of $2,856. Therefore, Tim's function has the smaller y-intercept. The answer is: Tim's with a y-intercept of $2,780.
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