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)
Find Tim's function equation using the point-slope form: y − 4580 = 225 ( x − 8 ) .
Calculate Tim's y-intercept by setting x = 0 : y = 2780 .
Calculate Paul's y-intercept from the given equation y − 1400 = 56 ( x + 26 ) by setting x = 0 : y = 2856 .
Compare the y-intercepts: Tim's y-intercept ($2,780) is smaller than Paul's y-intercept ($2,856), so the answer is Tim's with a y-intercept of $2 , 780 .
Explanation
Problem Analysis Let's analyze the problem. We need to find the y-intercepts of both Tim's and Paul's functions and compare them to determine which one is smaller.
Tim's Function Equation First, let's find the equation for Tim's function. We know the account increases at a rate of $225 per month, which means the slope of the line is 225. We also know that after 8 months, the account has $4,580. We can use the point-slope form of a linear equation: y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. In this case, m = 225 and ( x 1 , y 1 ) = ( 8 , 4580 ) . So the equation is:
y − 4580 = 225 ( x − 8 )
Tim's Y-Intercept To find the y-intercept of Tim's function, we set x = 0 :
y − 4580 = 225 ( 0 − 8 )
y − 4580 = − 1800
y = 4580 − 1800
y = 2780
So, the y-intercept of Tim's function is $2,780.
Paul's Function Equation Now, let's find the y-intercept of Paul's function. The equation is given as:
y − 1400 = 56 ( x + 26 )
Paul's Y-Intercept To find the y-intercept of Paul's function, we set x = 0 :
y − 1400 = 56 ( 0 + 26 )
y − 1400 = 56 ( 26 )
y − 1400 = 1456
y = 1456 + 1400
y = 2856
So, the y-intercept of Paul's function is $2,856.
Comparison and Conclusion Comparing the two y-intercepts, Tim's y-intercept is $2,780 and Paul's y-intercept is $2,856. Since 2780 < 2856, Tim's function has the smaller y-intercept.
Examples
Understanding y-intercepts is crucial in various real-life scenarios, such as analyzing business growth or predicting resource depletion. For instance, imagine a company's revenue can be modeled as a linear function of time. The y-intercept represents the initial revenue when the company started ( t = 0 ). By comparing the y-intercepts of different revenue models, you can quickly assess which company had a higher initial revenue. Similarly, in environmental science, if you're modeling the amount of a pollutant in a lake over time, the y-intercept indicates the initial level of pollution. Comparing y-intercepts helps determine which lake started with a higher pollution level, providing a baseline for further analysis and intervention strategies. In our problem, we found that Tim's savings account started with less money than Paul's, based on their respective linear models.
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 D.
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