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Chapter 3: Problem 70

The number \(V\) of computers infected by a computer virus increases according to the model \(V(t)=100 e^{4.6052 t}\), where \(t\) is the time in hours. Find the number of computers infected after (a) 1 hour, (b) 1.5 hours, and (c) 2 hours.

Short Answer

Expert verified
After 1 hour, approximately \[V(1) = 100e^{4.6052}\] computers are infected. After 1.5 hours, approximately \[V(1.5) = 100e^{6.9078}\] computers are infected. After 2 hours, approximately \[V(2) = 100e^{9.2104}\] computers are infected. Plug these values into a calculator to get the exact numbers.

Step by step solution

01

Understanding the Model

The model provided is \(V(t) = 100e^{4.6052t}\) which shows the number of computers \(V\) that will be infected by the virus at any given time \(t\) (in hours). Here, \(e\) is the base of the natural logarithm (approximately equal to 2.718).
02

Calculating for t = 1 hour

Substitute the time value \(t=1\) hour into the equation: \(V(1) = 100e^{4.6052 * 1}\). Calculate the exponent first and then multiply it with 100.
03

Calculating for t = 1.5 hours

Now substitute the time value \(t=1.5\) hours into the equation: \(V(1.5) = 100e^{4.6052 * 1.5}\). Again, calculate the exponent first and then multiply it with 100.
04

Calculating for t = 2 hours

Lastly, substitute the time value \(t=2\) hours into the equation: \(V(2) = 100e^{4.6052 * 2}\). Compute the exponent first and then multiply it with 100.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Modeling
Mathematical modeling is a powerful tool that helps us understand and predict real-world phenomena using mathematical concepts and structures.
In the given exercise, the model used is an exponential growth model, which describes the rate at which computer viruses spread. In this context:
  • The model is expressed as an equation: \( V(t) = 100e^{4.6052t} \), where \( V(t) \) represents the number of infected computers over time \( t \) in hours.
  • This model allows us to substitute different values of \( t \) to find how many computers will be infected at those specific times.
  • The constant 100 indicates the initial number of infected computers at \( t = 0 \), setting a baseline for measurements.
Through the model, insights can be gained about how quickly the infection spreads, helping to formulate strategies for intervention and control.
Natural Exponential Function
The natural exponential function plays a critical role in modeling exponential growth. It is represented by the equation \( e^x \), where \( e \) is Euler's number, approximately equal to 2.718. In the model \( V(t) = 100e^{4.6052t} \):
  • \( e^{4.6052t} \) is the component responsible for exponential growth, signifying that the number of infected computers grows rapidly over time.
  • The coefficient within the exponent, 4.6052, determines the growth rate, illustrating how quickly the number of infections accelerates.
The natural exponential function is widely used because of its unique property of consistent relative growth. This makes it ideal for calculations requiring growth rates, such as the spread of viruses and bacteria.
Problem Solving Steps
Approaching problems involving exponential growth requires clear steps.In our example problem, the task was to compute the number of infected computers for specific time periods using the given model:
  • **Step 1:** Identify the formula, \( V(t) = 100e^{4.6052t} \), representing the growth model. Recognize what each component, like \( e \) and 4.6052, represents.
  • **Step 2:** Substitute the given time values within the formula (i.e., \( t = 1 \), \( t = 1.5 \), and \( t = 2 \)). This practical step involves replacing \( t \) with the actual values you need to calculate.
  • **Step 3:** Calculate the expression. Start by computing the exponent first, \( 4.6052 imes t \), and then raise \( e \) to this power. Multiply the result by 100 to get the number of infected computers.
Following these steps allows for efficient and accurate problem solving, ensuring that results are obtained systematically and with clarity.

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Most popular questions from this chapter

The IQ scores for a sample of a class of returning adult students at a small northeastern college roughly follow the normal distribution \(y=0.0266 e^{-(x-100)^{2} / 450}, 70 \leq x \leq 115,\) where \(x\) is the IQ score. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average IQ score of an adult student.

Gaussian models are commonly used in probability and statistics to represent populations that are ________ ________.

At 8: 30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F}\), and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F}\). From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula is derived from a general cooling principle called Newton's Law of Cooling. It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death, and that the room temperature was a constant \(70^{\circ} \mathrm{F}\).) Use the formula to estimate the time of death of the person.

The amount of time (in hours per week) a student utilizes a math-tutoring center roughly follows the normal distribution \(y=0.7979 e^{-(x-5.4)^{2} / 0.5},\) \(4 \leq x \leq 7,\) where \(x\) is the number of hours. (a) Use a graphing utility to graph the function. (b) From the graph in part (a), estimate the average number of hours per week a student uses the tutoring center.

A laptop computer that costs $$\$ 1150$$ new has a book value of $$\$ 550$$ after 2 years. (a) Find the linear model \(V=m t+b\). (b) Find the exponential model \(V=a e^{k t}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.
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