Chapter ISG, Problem 7.9.3P

a.

To determine

To describe the methods of payment.

Answer to Problem 7.9.3P

In method 1 salary increases by $8 and in method 2 salary increases by 2percent

Explanation of Solution

Given information:

Months	Method 1 of payment	Method 2 of payment
1	$100	$0.01
2	$108	$0.02
3	$116	$0.04
4	$124	$0.08

Formula Used:

In arithmetic sequence the common difference is same.

The common difference is,

  $d=a2-a1$

In geometric sequence the common ratio is same.

The common ratio is,

  $r=\frac{a2}{a1}$

Calculation:

In method one payment, each month the salary increases by a common difference. The common difference is,

  $d=108-100$

On solving,

  $d=8$

Therefore, each month the salary increases by $8 in method 1. Hence, method 1 is a arithmetic progression.

In method two payment, each month the salary increases by a common ratio. The common ratio is,

  $r=\frac{0.02}{0.01}$

On solving,

  $r=2$

Therefore, each month the salary increases by 2 percent of the present salary in method 2. Hence, method 2 is a geometric progression

Conclusion:

In method 1 salary increases by $8 and in method 2 salary increases by 2percent

b.

To determine

To find a mathematic equation for both the methods.

Answer to Problem 7.9.3P

The function for method 1 is $y=8x+92$ and the function for method 2 is $y=0.01{(2)}^{x-1}$

Explanation of Solution

Given information:

Months	Method 1 of payment	Method 2 of payment
1	$100	$0.01
2	$108	$0.02
3	$116	$0.04
4	$124	$0.08

Common difference is 8

Common ratio is 2

Formula Used:

Arithmetic sequence is,

  $a_n=a+(n-1)d$

Geometric sequence is,

  $a_n=a{(r)}^{n-1}$

Calculation:

For method 1, substituting the values,

  $y=100+(x-1)8$

Opening the brackets,

  $y=100+8x-8$

On solving,

  $y=8x+92$

Hence, the function for method 1 is $y=8x+92$

For method 2, substituting the values,

  $y=0.01{(2)}^{x-1}$

Hence, the function for method 2 is $y=0.01{(2)}^{x-1}$

Conclusion:

The function for method 1 is $y=8x+92$ and the function for method 2 is $y=0.01{(2)}^{x-1}$

c.

To determine

To evaluate which method is better in 2 years.

Answer to Problem 7.9.3P

Method 2 is a better payment method.

Explanation of Solution

Given information:

The function for method 1 is $y=8x+92$

The function for method 2 is $y=0.01{(2)}^{x-1}$

Formula Used:

Equation substitution and solving.

Calculation:

1 year has 12 months. Therefore, 2 years has 24 months.

For method 1, substituting the values,

  $y=(8\times 24)+92$

On solving,

  $y=284$

Hence, the salary at the end of 2 years will be $284

For method 2, substituting the values,

  $y=0.01{(2)}^{24-1}$

So,

  $y=0.01{(2)}^{23}$

On solving,

  $y=83,886.08$

Hence, the salary at the end of 2years in method 2 will be $83886.08

Comparing the two salaries at the end of two years method 2 is better. Since the salary in method 2 is extremely high, than that of method 1, so method 2 will be better payment option.

Conclusion:

Method 2 is a better payment method.