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 1:**
Among 200 people, 150 either swim or jog or both. If 85 swim and 60 swim and jog, how many jog?
**Explanation:**
This problem involves a group of 200 people with overlapping activities of swimming and jogging. Out of these, 150 people participate in either swimming, jogging, or both activities. Specifically, 85 people swim, and among these, 60 people both swim and jog. The task is to determine the number of people who jog.
**Solution Approach:**
1. **Identify the Known Quantities:**
- Total people surveyed = 200
- Total people who swim or jog or both = 150
- People who swim = 85
- People who swim and jog = 60
2. **Use Set Notation and Formulas:**
- Let \( S \) represent swimmers, and \( J \) represent joggers.
- The intersection of these groups, \( S \cap J \), represents those who swim and jog, which is 60.
- We know from the problem that \( |S \cup J| = 150 \).
3. **Calculate the Number of Joggers:**
- Using the formula for union of two sets:
\[ |S \cup J| = |S| + |J| - |S \cap J| \]
- Substitute the known values:
\[ 150 = 85 + |J| - 60 \]
- Simplify to find \( |J| \):
\[ 150 = 85 + |J| - 60 \]
\[ 150 = 25 + |J| \]
\[ |J| = 125 \]
Thus, 125 people jog.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc28801d2-5244-43a8-94e3-88d6acce7d2a%2F41d39fe6-b4cf-4a6b-a849-cf6796c9ce51%2Fy6n7xfk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1:**
Among 200 people, 150 either swim or jog or both. If 85 swim and 60 swim and jog, how many jog?
**Explanation:**
This problem involves a group of 200 people with overlapping activities of swimming and jogging. Out of these, 150 people participate in either swimming, jogging, or both activities. Specifically, 85 people swim, and among these, 60 people both swim and jog. The task is to determine the number of people who jog.
**Solution Approach:**
1. **Identify the Known Quantities:**
- Total people surveyed = 200
- Total people who swim or jog or both = 150
- People who swim = 85
- People who swim and jog = 60
2. **Use Set Notation and Formulas:**
- Let \( S \) represent swimmers, and \( J \) represent joggers.
- The intersection of these groups, \( S \cap J \), represents those who swim and jog, which is 60.
- We know from the problem that \( |S \cup J| = 150 \).
3. **Calculate the Number of Joggers:**
- Using the formula for union of two sets:
\[ |S \cup J| = |S| + |J| - |S \cap J| \]
- Substitute the known values:
\[ 150 = 85 + |J| - 60 \]
- Simplify to find \( |J| \):
\[ 150 = 85 + |J| - 60 \]
\[ 150 = 25 + |J| \]
\[ |J| = 125 \]
Thus, 125 people jog.
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