Question
Carmen is considering two plans to pay off a $10,000 loan . The table show the amount remaining on the loan after x years . Which plan should carmen use to pay off the loan as soon as possible ? Justify your answer using a function model ?
Hint:
1. When the difference between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant. i.e. y(n)- y(n-1) is constant for any value of n, the function is known as a linear function.
2. When the difference between 2 consecutive differences for output values (y values) for a given constant change in the input values (x values) is constant. i.e. dy(n)- dy(n-1) is constant for any value of n, the function is known as a quadratic function.
3. When the ratio between 2 consecutive output values (y values) for a given constant change in the input values (x values) is constant i.e. y(n)/y(n-1) is constant for any value of n, the function is known as an exponential function.
The correct answer is: Carmen should use Plan B to pay off the loan as soon as possible.
Step-by-step solution:-
From the given table, we observe the following readings-
x1 = 0, y1(A) = 10,000, y1(B) = 10,000;
x2 = 1, y2(A) = 9,000, y1(B) = 9,500;
x3 = 2, y3(A) = 8,100, y1(B) = 9,000;
x4 = 3, y4(A) = 7,290, y1(B) = 8,500
x5 = 4, y5(A) = 6,561, y5(B) = 8,000
Difference between 2 consecutive x values-
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
dx4 = x5 - x4 = 4 - 3 = 1
a). For Plan A-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 9,000 - 10,000 = -1,000
dy2 = y3 - y2 = 8,100 - 9,000 = -900
dy3 = y4 - y3 = 7,290 - 8,100 = -810
dy4 = y5 - y4 = 6,561 - 7,290 = -729
We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
Hence, the given function is not a linear function.
Now, ratio between 2 consecutive y values-
y2/y1 = 9,000/10,000 = 0.9
y3/y2 = 8,100/9,000 = 0.9
y4/y3 = 7,290/8,100 = 0.9
y5/y4 = 6,561/7,290 = 0.9
We observe that difference between 2 consecutive y values is constant i.e. 0.9. Hence, the given function is an exponential function.
b). For Plan B-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 9,500 - 10,000 = -500
dy2 = y3 - y2 = 9,000 - 9,500 = -500
dy3 = y4 - y3 = 8,500 - 9,000 = -500
dy4 = y5 - y4 = 8,000 - 8,500 = -500
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. -500.
Hence, the given function is a linear function.
We know from the above calculations that the loan repayment as per Plan A is an exponential function and that for Plan B is a Linear function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing/decreasing at an exponential level. Hence, the loan repayment as per Plan A will be slower as compared to that in Plan B because the amount paid each month keeps decreasing at an exponential level for Plan A.
Hence, loan payment as per Plan B will be faster.
Final Answer:-
∴ Carmen should use Plan B to pay off the loan as soon as possible.
x2 = 1, y2(A) = 9,000, y1(B) = 9,500;
x3 = 2, y3(A) = 8,100, y1(B) = 9,000;
x4 = 3, y4(A) = 7,290, y1(B) = 8,500
x5 = 4, y5(A) = 6,561, y5(B) = 8,000
dx1 = x2 - x1 = 1 - 0 = 1
dx2 = x3 - x2 = 2 - 1 = 1
dx3 = x4 - x3 = 3 - 2 = 1
dx4 = x5 - x4 = 4 - 3 = 1
a). For Plan A-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 9,000 - 10,000 = -1,000
dy2 = y3 - y2 = 8,100 - 9,000 = -900
dy3 = y4 - y3 = 7,290 - 8,100 = -810
dy4 = y5 - y4 = 6,561 - 7,290 = -729
We observe that the difference for every consecutive x values is constant i.e. 1 but for y values the difference is not constant.
Hence, the given function is not a linear function.
Now, ratio between 2 consecutive y values-
y2/y1 = 9,000/10,000 = 0.9
y3/y2 = 8,100/9,000 = 0.9
y4/y3 = 7,290/8,100 = 0.9
y5/y4 = 6,561/7,290 = 0.9
We observe that difference between 2 consecutive y values is constant i.e. 0.9. Hence, the given function is an exponential function.
b). For Plan B-
Difference between 2 consecutive y values-
dy1 = y2 - y1 = 9,500 - 10,000 = -500
dy2 = y3 - y2 = 9,000 - 9,500 = -500
dy3 = y4 - y3 = 8,500 - 9,000 = -500
dy4 = y5 - y4 = 8,000 - 8,500 = -500
We observe that the difference for every consecutive x values is constant i.e. 1 and for y values the difference is constant i.e. -500.
Hence, the given function is a linear function.
We know from the above calculations that the loan repayment as per Plan A is an exponential function and that for Plan B is a Linear function. We also know that for a linear function, the rate of change in output for a given change in input is linear i.e. constant. However, for an exponential function, the rate of change in output for a given change in input keeps increasing/decreasing at an exponential level. Hence, the loan repayment as per Plan A will be slower as compared to that in Plan B because the amount paid each month keeps decreasing at an exponential level for Plan A.
Hence, loan payment as per Plan B will be faster.
Final Answer:-
∴ Carmen should use Plan B to pay off the loan as soon as possible.
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