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The average annual salary of the employees of a company in the year 2005 was [tex][tex]$\$[/tex]70,000[tex]$[/tex]. It increased by the same factor each year, and in 2006, the average annual salary was [tex]$[/tex]\[tex]$82,000$[/tex][/tex]. Let [tex][tex]$y$[/tex][/tex] represent the average annual salary, in thousand dollars, after [tex][tex]$x$[/tex][/tex] years since 2005. Which of the following best represents the relationship between [tex][tex]$x$[/tex][/tex] and [tex][tex]$y$[/tex][/tex]?

A. [tex][tex]$y=70(1.17)^x$[/tex][/tex]
B. [tex][tex]$y=82(1.17)^x$[/tex][/tex]
C. [tex][tex]$y=70(2.2)^x$[/tex][/tex]
D. [tex][tex]$y=82(2.2)$[/tex][/tex]

Because of the rainy season, the depth in a pond increases [tex][tex]$3\%$[/tex][/tex] each week. Before the rainy season started, the pond was 10 feet deep. What is the equation that best represents the depth of the pond each week?

Hint: Use the formula [tex][tex]$y=P(1+r)^x$[/tex][/tex].



Answer :

GinnyAnswer
Certainly! Let's work step-by-step through the problem involving the depth of the pond increasing each week due to the rainy season.

### Problem Statement

Due to the rainy season, the depth in a pond increases by 3% each week. Before the rainy season started, the pond was 10 feet deep. We are looking for an equation that best represents the depth of the pond each week.

### Given Information

- Initial depth of the pond [tex]\( P = 10 \)[/tex] feet
- Weekly increase rate [tex]\( r = 3\% = 0.03 \)[/tex]

### Formula to Use

The appropriate formula to represent exponential growth is:
[tex]\[ y = P (1 + r)^x \][/tex]

Where:
- [tex]\( y \)[/tex] is the depth of the pond after [tex]\( x \)[/tex] weeks.
- [tex]\( P \)[/tex] is the initial depth of the pond.
- [tex]\( r \)[/tex] is the rate of increase per period (week, in this case).
- [tex]\( x \)[/tex] is the number of weeks.

### Step-by-Step Solution

1. Initial Depth:
The initial depth [tex]\( P \)[/tex] is given to be 10 feet.

2. Weekly Increase Rate:
The depth of the pond increases by 3% each week which translates to [tex]\( r = 0.03 \)[/tex].

3. Equation for Depth After [tex]\( x \)[/tex] Weeks:
By substituting [tex]\( P = 10 \)[/tex] feet and [tex]\( r = 0.03 \)[/tex] into the general formula:
[tex]\[ y = 10 (1 + 0.03)^x \][/tex]
Simplifying inside the parentheses:
[tex]\[ y = 10 (1.03)^x \][/tex]

Therefore, the equation that best represents the depth of the pond each week is:
[tex]\[ y = 10 (1.03)^x \][/tex]

### Verification with Given Data

Let's verify by calculating the depths for few different weeks:

1. After 1 Week:
[tex]\[ y = 10 \times (1.03)^1 = 10 \times 1.03 = 10.3 \text{ feet} \][/tex]

2. After 2 Weeks:
[tex]\[ y = 10 \times (1.03)^2 = 10 \times 1.0609 = 10.609 \text{ feet} \][/tex]

3. After 4 Weeks:
[tex]\[ y = 10 \times (1.03)^4 = 10 \times 1.12550881 = 11.2550881 \text{ feet} \][/tex]

These results verify that the provided equation [tex]\( y = 10 (1.03)^x \)[/tex] correctly represents the increase in depth of the pond due to the rainy season.

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