Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

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].

Sagot :

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.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.