Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find the distance between the points (3, 8) and (-1, 11) and the slope of the line containing them.
**Solution Explanation:**
To solve this problem, we need to calculate two values: the distance between the points and the slope of the line that passes through them. Here’s how to find each:
1. **Distance between Two Points:**
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the provided points \((3, 8)\) and \((-1, 11)\), we have:
\[
d = \sqrt{((-1) - 3)^2 + (11 - 8)^2}
\]
\[
d = \sqrt{(-4)^2 + 3^2}
\]
\[
d = \sqrt{16 + 9}
\]
\[
d = \sqrt{25} = 5
\]
Thus, the distance between the points is 5 units.
2. **Slope of the Line:**
The slope \(m\) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using our points \((3, 8)\) and \((-1, 11)\), we can substitute the values:
\[
m = \frac{11 - 8}{-1 - 3}
\]
\[
m = \frac{3}{-4}
\]
Therefore, the slope of the line is \(-\frac{3}{4}\).
By following these calculations, you can obtain both the distance and the slope for these two points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b3925e9-3698-4de9-9f01-eff4f69c62e5%2Fb61ea42a-817e-4f05-a44f-42ff9bace47c%2F0xmdirf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the distance between the points (3, 8) and (-1, 11) and the slope of the line containing them.
**Solution Explanation:**
To solve this problem, we need to calculate two values: the distance between the points and the slope of the line that passes through them. Here’s how to find each:
1. **Distance between Two Points:**
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Plugging in the provided points \((3, 8)\) and \((-1, 11)\), we have:
\[
d = \sqrt{((-1) - 3)^2 + (11 - 8)^2}
\]
\[
d = \sqrt{(-4)^2 + 3^2}
\]
\[
d = \sqrt{16 + 9}
\]
\[
d = \sqrt{25} = 5
\]
Thus, the distance between the points is 5 units.
2. **Slope of the Line:**
The slope \(m\) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as follows:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using our points \((3, 8)\) and \((-1, 11)\), we can substitute the values:
\[
m = \frac{11 - 8}{-1 - 3}
\]
\[
m = \frac{3}{-4}
\]
Therefore, the slope of the line is \(-\frac{3}{4}\).
By following these calculations, you can obtain both the distance and the slope for these two points.
Expert Solution
Step 1
We need to find the distance between (3, 8) and (-1, 11).
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