Question

1. c. if mr. robert’s had signed a 5-year contract with increase of 2.45% what will be his annual salary at the end of the 5 years?

Explanation:

To solve this problem, we need to know Mr. Robert’s initial annual salary. Let's assume his initial annual salary is \( P \). The formula for compound growth (since the salary increase is compounded annually at a rate of \( r = 2.45\% = 0.0245 \) over \( n = 5 \) years) is:

\[ A = P(1 + r)^n \]

Step 1: Identify the values

Let's assume Mr. Robert’s initial salary \( P \) (we need this value, but since it's not provided, let's assume an example. Let's say \( P = \$100,000 \) for illustration). The rate \( r = 0.0245 \) and the number of years \( n = 5 \).

Step 2: Substitute the values into the formula

\[ A = 100000(1 + 0.0245)^5 \]

Step 3: Calculate \( (1 + 0.0245)^5 \)

First, calculate \( 1 + 0.0245 = 1.0245 \). Then, \( 1.0245^5 \approx 1.0245 \times 1.0245 \times 1.0245 \times 1.0245 \times 1.0245 \approx 1.1293 \) (using a calculator for more precision: \( 1.0245^5 \approx e^{5 \times \ln(1.0245)} \approx e^{5 \times 0.0242} \approx e^{0.121} \approx 1.1287 \) or directly calculating \( 1.0245^5 \approx 1.1287 \)).

Step 4: Calculate the final amount \( A \)

\[ A = 100000 \times 1.1287 \approx \$112,870 \]

If we use a more precise calculation of \( 1.0245^5 \):

\[ 1.0245^5 = 1.0245 \times 1.0245 = 1.04960025 \]
\[ 1.04960025 \times 1.0245 \approx 1.07532546 \]
\[ 1.07532546 \times 1.0245 \approx 1.10167043 \]
\[ 1.10167043 \times 1.0245 \approx 1.1287 \]

So, \( A \approx 100000 \times 1.1287 = 112870 \).

If the initial salary \( P \) is different, say \( P = \$50,000 \):

\[ A = 50000(1.0245)^5 \approx 50000 \times 1.1287 = \$56,435 \]

Since the initial salary is not provided in the problem, we need that information. Let's assume the initial salary is \( P \). Then the formula to find the salary after 5 years is:

\[ A = P(1 + 0.0245)^5 \approx P \times 1.1287 \]

If we take a common initial salary, for example, if Mr. Robert’s initial salary is \( \$80,000 \):

\[ A = 80000 \times 1.1287 \approx \$90,296 \]

Final Answer

To get the exact answer, we need the initial salary \( P \). Using the formula \( A = P(1 + 0.0245)^5 \), substitute the value of \( P \) (initial salary) to find the annual salary at the end of 5 years. For example, if \( P = \$100,000 \), the annual salary at the end of 5 years is approximately \(\$112,870\) (this value will change based on the actual initial salary).

Answer:

To solve this problem, we need to know Mr. Robert’s initial annual salary. Let's assume his initial annual salary is \( P \). The formula for compound growth (since the salary increase is compounded annually at a rate of \( r = 2.45\% = 0.0245 \) over \( n = 5 \) years) is:

\[ A = P(1 + r)^n \]

Step 1: Identify the values

Let's assume Mr. Robert’s initial salary \( P \) (we need this value, but since it's not provided, let's assume an example. Let's say \( P = \$100,000 \) for illustration). The rate \( r = 0.0245 \) and the number of years \( n = 5 \).

Step 2: Substitute the values into the formula

\[ A = 100000(1 + 0.0245)^5 \]

Step 3: Calculate \( (1 + 0.0245)^5 \)

First, calculate \( 1 + 0.0245 = 1.0245 \). Then, \( 1.0245^5 \approx 1.0245 \times 1.0245 \times 1.0245 \times 1.0245 \times 1.0245 \approx 1.1293 \) (using a calculator for more precision: \( 1.0245^5 \approx e^{5 \times \ln(1.0245)} \approx e^{5 \times 0.0242} \approx e^{0.121} \approx 1.1287 \) or directly calculating \( 1.0245^5 \approx 1.1287 \)).

Step 4: Calculate the final amount \( A \)

\[ A = 100000 \times 1.1287 \approx \$112,870 \]

If we use a more precise calculation of \( 1.0245^5 \):

\[ 1.0245^5 = 1.0245 \times 1.0245 = 1.04960025 \]
\[ 1.04960025 \times 1.0245 \approx 1.07532546 \]
\[ 1.07532546 \times 1.0245 \approx 1.10167043 \]
\[ 1.10167043 \times 1.0245 \approx 1.1287 \]

So, \( A \approx 100000 \times 1.1287 = 112870 \).

If the initial salary \( P \) is different, say \( P = \$50,000 \):

\[ A = 50000(1.0245)^5 \approx 50000 \times 1.1287 = \$56,435 \]

Since the initial salary is not provided in the problem, we need that information. Let's assume the initial salary is \( P \). Then the formula to find the salary after 5 years is:

\[ A = P(1 + 0.0245)^5 \approx P \times 1.1287 \]

If we take a common initial salary, for example, if Mr. Robert’s initial salary is \( \$80,000 \):

\[ A = 80000 \times 1.1287 \approx \$90,296 \]

Final Answer

To get the exact answer, we need the initial salary \( P \). Using the formula \( A = P(1 + 0.0245)^5 \), substitute the value of \( P \) (initial salary) to find the annual salary at the end of 5 years. For example, if \( P = \$100,000 \), the annual salary at the end of 5 years is approximately \(\$112,870\) (this value will change based on the actual initial salary).