**Educational Website Content****Title:** Calculation of Virus Spread Over Time**Description:**Understanding how a virus spreads through a population is crucial for controlling outbreaks. In mathematical terms, the rate at which a virus spreads can be represented by a specific function. Below, we explore a function that describes this rate and calculate the number of people infected over a given time period.**Mathematical Model:**The rate at which a virus spreads through a population (measured in people per day) is given by the function:\[ r(t) = 2.33e^{1.165t} \]where:- $ r(t) $ is the rate of infection (people per day)- $ t $ is the number of days since the initial infection**Problem Statement:**Calculate the number of people infected between days 7 and 14. Round your answer to the nearest person.**Detailed Explanation:**To find out how many people were infected between days 7 and 14, we need to integrate the given function $ r(t) $ from $ t = 7 $ to $ t = 14 $.**Steps:**1. Set up the integral to represent the total number of infections between days 7 and 14:\[ \int_{7}^{14} 2.33e^{1.165t} \, dt \]2. Solve the integral analytically:\[ \int 2.33e^{1.165t} \, dt = \frac{2.33}{1.165} e^{1.165t} + C \]\[ = 2.00 e^{1.165t} + C \]3. Evaluate the integral from $ t = 7 $ to $ t = 14 $:\[ \left[ 2.00 e^{1.165t} \right]_{7}^{14} = 2.00 e^{1.165 \cdot 14} - 2.00 e^{1.165 \cdot 7} \]4. Calculate the exponential values and subtract:\[ 2.00 e^{1.165 \cdot 14} - 2.00 e^{1.165 \cdot 7} \approx 2.00 (1235072.36) - 2.00 (518.02) \]\[ \approx 2470144 **How much work does it take this squirrel to suck all the beer out of one of these cylindrical cans?** *Note: The straw reaches 3 in out of the can, the can is 4.5 in tall with a diameter of 2.5 in. This beer weighs 0.036 lb/in³.*

The rate at which a virus spreads through a population (measured in people per day) is given by the function r(t) = 2.33e1.165t, where t is the number of days since the initial infection. How many people were infected between days 7 and 14? Round your answer to the nearest person.

The rate at which a virus spreads through a population (measured in people per day) is given by the function r(t) = 2.33e1.165t, where t is the number of days since the initial infection. How many people were infected between days 7 and 14? Round your answer to the nearest person.

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Educational Website Content**

**Title:** Calculation of Virus Spread Over Time

**Description:**
Understanding how a virus spreads through a population is crucial for controlling outbreaks. In mathematical terms, the rate at which a virus spreads can be represented by a specific function. Below, we explore a function that describes this rate and calculate the number of people infected over a given time period.

**Mathematical Model:**
The rate at which a virus spreads through a population (measured in people per day) is given by the function:

\[ r(t) = 2.33e^{1.165t} \]

where:
- $ r(t) $ is the rate of infection (people per day)
- $ t $ is the number of days since the initial infection

**Problem Statement:**
Calculate the number of people infected between days 7 and 14. Round your answer to the nearest person.

**Detailed Explanation:**
To find out how many people were infected between days 7 and 14, we need to integrate the given function $ r(t) $ from $ t = 7 $ to $ t = 14 $.

**Steps:**
1. Set up the integral to represent the total number of infections between days 7 and 14:

\[ \int_{7}^{14} 2.33e^{1.165t} \, dt \]

2. Solve the integral analytically:

\[ \int 2.33e^{1.165t} \, dt = \frac{2.33}{1.165} e^{1.165t} + C \]
\[ = 2.00 e^{1.165t} + C \]

3. Evaluate the integral from $ t = 7 $ to $ t = 14 $:

\[ \left[ 2.00 e^{1.165t} \right]_{7}^{14} = 2.00 e^{1.165 \cdot 14} - 2.00 e^{1.165 \cdot 7} \]

4. Calculate the exponential values and subtract:

\[ 2.00 e^{1.165 \cdot 14} - 2.00 e^{1.165 \cdot 7} \approx 2.00 (1235072.36) - 2.00 (518.02) \]
\[ \approx 2470144

**How much work does it take this squirrel to suck all the beer out of one of these cylindrical cans?**

*Note: The straw reaches 3 in out of the can, the can is 4.5 in tall with a diameter of 2.5 in. This beer weighs 0.036 lb/in³.*

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