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-

x_{1} = 0, y_{1}(A) = 10,000, y_{1}(B) = 10,000;

x_{2} = 1, y_{2}(A) = 9,000, y_{1}(B) = 9,500;

x_{3} = 2, y_{3}(A) = 8,100, y_{1}(B) = 9,000;

x_{4} = 3, y_{4}(A) = 7,290, y_{1}(B) = 8,500

x_{5} = 4, y_{5}(A) = 6,561, y_{5}(B) = 8,000

Difference between 2 consecutive x values-

dx_{1} = x_{2} - x_{1} = 1 - 0 = 1

dx_{2} = x_{3} - x_{2} = 2 - 1 = 1

dx_{3} = x_{4} - x_{3} = 3 - 2 = 1

dx_{4} = x_{5} - x_{4} = 4 - 3 = 1

a). For Plan A-

Difference between 2 consecutive y values-

dy_{1} = y_{2} - y_{1} = 9,000 - 10,000 = -1,000

dy_{2} = y_{3} - y_{2} = 8,100 - 9,000 = -900

dy_{3} = y_{4} - y_{3} = 7,290 - 8,100 = -810

dy_{4} = y_{5} - y_{4} = 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-

y_{2}/y_{1} = 9,000/10,000 = 0.9

y_{3}/y_{2} = 8,100/9,000 = 0.9

y_{4}/y_{3} = 7,290/8,100 = 0.9

y_{5}/y_{4} = 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-

dy_{1} = y_{2} - y_{1} = 9,500 - 10,000 = -500

dy_{2} = y_{3} - y_{2} = 9,000 - 9,500 = -500

dy_{3} = y_{4} - y_{3} = 8,500 - 9,000 = -500

dy_{4} = y_{5} - y_{4} = 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.

_{1}= 0, y

_{1}(A) = 10,000, y

_{1}(B) = 10,000;

x

_{2}= 1, y

_{2}(A) = 9,000, y

_{1}(B) = 9,500;

x

_{3}= 2, y

_{3}(A) = 8,100, y

_{1}(B) = 9,000;

x

_{4}= 3, y

_{4}(A) = 7,290, y

_{1}(B) = 8,500

x

_{5}= 4, y

_{5}(A) = 6,561, y

_{5}(B) = 8,000

dx

_{1}= x

_{2}- x

_{1}= 1 - 0 = 1

dx

_{2}= x

_{3}- x

_{2}= 2 - 1 = 1

dx

_{3}= x

_{4}- x

_{3}= 3 - 2 = 1

dx

_{4}= x

_{5}- x

_{4}= 4 - 3 = 1

a). For Plan A-

Difference between 2 consecutive y values-

dy

_{1}= y

_{2}- y

_{1}= 9,000 - 10,000 = -1,000

dy

_{2}= y

_{3}- y

_{2}= 8,100 - 9,000 = -900

dy

_{3}= y

_{4}- y

_{3}= 7,290 - 8,100 = -810

dy

_{4}= y

_{5}- y

_{4}= 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-

y

_{2}/y

_{1}= 9,000/10,000 = 0.9

y

_{3}/y

_{2}= 8,100/9,000 = 0.9

y

_{4}/y

_{3}= 7,290/8,100 = 0.9

y

_{5}/y

_{4}= 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-

dy

_{1}= y

_{2}- y

_{1}= 9,500 - 10,000 = -500

dy

_{2}= y

_{3}- y

_{2}= 9,000 - 9,500 = -500

dy

_{3}= y

_{4}- y

_{3}= 8,500 - 9,000 = -500

dy

_{4}= y

_{5}- y

_{4}= 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.