Solve for the slope, y-intercept, and equation of the trend line for the following the data set. Show your work and explain the steps you used to solve. Round your answer to the nearest hundredth. Concentration (M) Absorbance 0.00 0.000 0.179 0.444 0.20 0.40
Solve for the slope, y-intercept, and equation of the trend line for the following the data set. Show your work and explain the steps you used to solve. Round your answer to the nearest hundredth. Concentration (M) Absorbance 0.00 0.000 0.179 0.444 0.20 0.40
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![## Experimental Data: Absorbance vs. Concentration
In this experiment, we measured the absorbance of a solution at various concentrations. The data collected is tabulated below.
| **Concentration (M)** | **Absorbance** |
|-----------------------|----------------|
| 0.00 | 0.000 |
| 0.20 | 0.179 |
| 0.40 | 0.444 |
| 0.60 | 0.654 |
| 0.80 | 0.898 |
### Explanation of Data
- **Concentration (M)**: This column represents the molar concentration of the solution.
- **Absorbance**: This column represents the absorbance values recorded at each corresponding concentration.
### Understanding the Data
This data can be used to analyze the relationship between concentration and absorbance of the solution. According to Beer-Lambert Law, absorbance (A) is directly proportional to the concentration (C) of the solution, given by the equation:
\[ A = \epsilon \cdot l \cdot C \]
where \( \epsilon \) is the molar absorptivity, and \( l \) is the path length of the cell. By plotting absorbance against concentration, we can determine the molar absorptivity and validate the linear relationship between these two parameters.
For educational purposes, it is important to see how increasing concentration affects the absorbance readings, as higher concentrations typically result in higher absorbance values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4056cef-3867-489f-a260-f13d42f9bf0e%2Fc4558b2e-dde5-498d-8801-c4bd8c9f840b%2Ffckp8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Experimental Data: Absorbance vs. Concentration
In this experiment, we measured the absorbance of a solution at various concentrations. The data collected is tabulated below.
| **Concentration (M)** | **Absorbance** |
|-----------------------|----------------|
| 0.00 | 0.000 |
| 0.20 | 0.179 |
| 0.40 | 0.444 |
| 0.60 | 0.654 |
| 0.80 | 0.898 |
### Explanation of Data
- **Concentration (M)**: This column represents the molar concentration of the solution.
- **Absorbance**: This column represents the absorbance values recorded at each corresponding concentration.
### Understanding the Data
This data can be used to analyze the relationship between concentration and absorbance of the solution. According to Beer-Lambert Law, absorbance (A) is directly proportional to the concentration (C) of the solution, given by the equation:
\[ A = \epsilon \cdot l \cdot C \]
where \( \epsilon \) is the molar absorptivity, and \( l \) is the path length of the cell. By plotting absorbance against concentration, we can determine the molar absorptivity and validate the linear relationship between these two parameters.
For educational purposes, it is important to see how increasing concentration affects the absorbance readings, as higher concentrations typically result in higher absorbance values.
![### Solving for the Slope, Y-intercept, and Equation of the Trend Line
In this activity, we will solve for the slope, y-intercept, and the equation of the trend line for the provided data set. We will also explain each step used in the calculations. Once completed, round your answers to the nearest hundredth.
#### Data Set
Here is the data set for concentration and absorbance:
| Concentration (M) | Absorbance |
|-------------------|------------|
| 0.00 | 0.000 |
| 0.20 | 0.179 |
| 0.40 | 0.444 |
#### Steps to Solve:
1. **Determine the Slope (m):**
The slope of the trend line can be determined using the formula:
\[
m = \frac{(N \sum xy - \sum x \sum y)}{(N \sum x^2 - (\sum x)^2)}
\]
where \(N\) is the number of data points.
For our data:
- \(N = 3\)
- \(\sum x = 0.00 + 0.20 + 0.40 = 0.60\)
- \(\sum y = 0.000 + 0.179 + 0.444 = 0.623\)
- \(\sum xy = (0.00 \cdot 0.000) + (0.20 \cdot 0.179) + (0.40 \cdot 0.444) = 0.00 + 0.0358 + 0.1776 = 0.2134\)
- \(\sum x^2 = (0.00^2) + (0.20^2) + (0.40^2) = 0.00 + 0.04 + 0.16 = 0.20\)
Plugging these values into the slope formula:
\[
m = \frac{(3 \cdot 0.2134 - 0.60 \cdot 0.623)}{(3 \cdot 0.20 - (0.60)^2)}
\]
\[
m = \frac{(0.6402 - 0.373](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4056cef-3867-489f-a260-f13d42f9bf0e%2Fc4558b2e-dde5-498d-8801-c4bd8c9f840b%2Fafy0r1p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for the Slope, Y-intercept, and Equation of the Trend Line
In this activity, we will solve for the slope, y-intercept, and the equation of the trend line for the provided data set. We will also explain each step used in the calculations. Once completed, round your answers to the nearest hundredth.
#### Data Set
Here is the data set for concentration and absorbance:
| Concentration (M) | Absorbance |
|-------------------|------------|
| 0.00 | 0.000 |
| 0.20 | 0.179 |
| 0.40 | 0.444 |
#### Steps to Solve:
1. **Determine the Slope (m):**
The slope of the trend line can be determined using the formula:
\[
m = \frac{(N \sum xy - \sum x \sum y)}{(N \sum x^2 - (\sum x)^2)}
\]
where \(N\) is the number of data points.
For our data:
- \(N = 3\)
- \(\sum x = 0.00 + 0.20 + 0.40 = 0.60\)
- \(\sum y = 0.000 + 0.179 + 0.444 = 0.623\)
- \(\sum xy = (0.00 \cdot 0.000) + (0.20 \cdot 0.179) + (0.40 \cdot 0.444) = 0.00 + 0.0358 + 0.1776 = 0.2134\)
- \(\sum x^2 = (0.00^2) + (0.20^2) + (0.40^2) = 0.00 + 0.04 + 0.16 = 0.20\)
Plugging these values into the slope formula:
\[
m = \frac{(3 \cdot 0.2134 - 0.60 \cdot 0.623)}{(3 \cdot 0.20 - (0.60)^2)}
\]
\[
m = \frac{(0.6402 - 0.373
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