Identify The Initial Amount $a$ And The Rate Of Decay $r$ (as A Percent) Of The Exponential Function $g(t) = 240(0.75)^t$. Evaluate The Function When $t = 3$. Round Your Answer To The Nearest Tenth.$a =

Introduction to Exponential Decay

Exponential decay functions are a type of mathematical function that describes how a quantity decreases over time. These functions are characterized by a constant rate of decay, which is a key factor in determining the behavior of the function. In this article, we will focus on identifying the initial amount and the rate of decay of an exponential function, and then evaluate the function at a specific time.

The Exponential Function

The exponential function we will be working with is given by the equation:

$$g(t) = 240(0.75)^t$$

where $t$ is the time variable, and $g(t)$ is the value of the function at time $t$. The initial amount $a$ is the value of the function when $t = 0$, and the rate of decay $r$ is the constant rate at which the function decreases over time.

Identifying the Initial Amount

To identify the initial amount $a$, we need to find the value of the function when $t = 0$. We can do this by substituting $t = 0$ into the equation:

$$g(0) = 240(0.75)^0$$

Since any number raised to the power of 0 is equal to 1, we have:

$$g(0) = 240(1) = 240$$

Therefore, the initial amount $a$ is equal to 240.

Identifying the Rate of Decay

To identify the rate of decay $r$, we need to find the constant rate at which the function decreases over time. We can do this by examining the coefficient of the exponential term:

$$g(t) = 240(0.75)^t$$

The coefficient of the exponential term is 0.75, which is less than 1. This means that the function decreases over time, and the rate of decay is given by the value of the coefficient.

To express the rate of decay as a percent, we can multiply the coefficient by 100:

$$r = 0.75 \times 100 = 75\%$$

Therefore, the rate of decay $r$ is equal to 75%.

Evaluating the Function at $t = 3$

Now that we have identified the initial amount and the rate of decay, we can evaluate the function at $t = 3$. We can do this by substituting $t = 3$ into the equation:

$$g(3) = 240(0.75)^3$$

Using a calculator, we can evaluate the expression:

$$g(3) = 240(0.75)^3 = 240 \times 0.421875 = 100.6875$$

Rounding this value to the nearest tenth, we get:

$$g(3) = 100.7$$

Therefore, the value of the function at $t = 3$ is approximately 100.7.

Conclusion

In this article, we identified the initial amount and the rate of decay of an exponential function, and then evaluated the function at a specific time. We found that the initial amount $a$ is equal to 240, and the rate of decay $r$ is equal to 75%. We also found that the value of the function at $t = 3$ is approximately 100.7. This demonstrates the importance of understanding exponential decay functions in mathematics and their applications in real-world problems.

Exercises

1. Find the value of the function when $t = 2$.
2. Find the rate of decay of the function when $t = 5$.
3. Evaluate the function at $t = 10$.

Answers

1. $g(2) = 240(0.75)^2 = 180$
2. $r = 0.75 \times 100 = 75\%$
3. $g(10) = 240(0.75)^{10} = 12.5$

References

• [1] "Exponential Decay Functions" by Math Is Fun
• [2] "Exponential Functions" by Khan Academy
  Exponential Decay Functions: Q&A =====================================

Introduction

Exponential decay functions are a type of mathematical function that describes how a quantity decreases over time. These functions are characterized by a constant rate of decay, which is a key factor in determining the behavior of the function. In this article, we will answer some common questions about exponential decay functions.

Q: What is an exponential decay function?

A: An exponential decay function is a type of mathematical function that describes how a quantity decreases over time. It is characterized by a constant rate of decay, which is a key factor in determining the behavior of the function.

Q: How do I identify the initial amount and the rate of decay of an exponential decay function?

A: To identify the initial amount and the rate of decay of an exponential decay function, you need to examine the equation of the function. The initial amount is the value of the function when t = 0, and the rate of decay is the constant rate at which the function decreases over time.

Q: What is the formula for an exponential decay function?

A: The formula for an exponential decay function is:

g(t) = a(1 - r)^t

where g(t) is the value of the function at time t, a is the initial amount, r is the rate of decay, and t is the time variable.

Q: How do I evaluate an exponential decay function at a specific time?

A: To evaluate an exponential decay function at a specific time, you need to substitute the value of t into the equation of the function. For example, if you want to evaluate the function g(t) = 240(0.75)^t at t = 3, you would substitute t = 3 into the equation:

g(3) = 240(0.75)^3

Q: What is the significance of the rate of decay in an exponential decay function?

A: The rate of decay is a key factor in determining the behavior of an exponential decay function. It determines how quickly the function decreases over time. A higher rate of decay means that the function decreases more quickly, while a lower rate of decay means that the function decreases more slowly.

Q: Can I use exponential decay functions to model real-world problems?

A: Yes, exponential decay functions can be used to model a wide range of real-world problems, including population growth and decline, radioactive decay, and chemical reactions.

Q: How do I graph an exponential decay function?

A: To graph an exponential decay function, you need to plot the values of the function at different times. You can use a graphing calculator or a computer program to graph the function.

Q: What are some common applications of exponential decay functions?

A: Exponential decay functions have a wide range of applications, including:

• Modeling population growth and decline
• Describing radioactive decay
• Modeling chemical reactions
• Analyzing financial data

Q: Can I use exponential decay functions to solve problems in other fields?

A: Yes, exponential decay functions can be used to solve problems in a wide range of fields, including physics, engineering, economics, and biology.

Conclusion

In this article, we have answered some common questions about exponential decay functions. We have discussed the formula for an exponential decay function, how to identify the initial amount and the rate of decay, and how to evaluate the function at a specific time. We have also discussed the significance of the rate of decay and some common applications of exponential decay functions.

Exercises

1. Find the value of the function g(t) = 240(0.75)^t at t = 5.
2. Find the rate of decay of the function g(t) = 120(0.9)^t.
3. Evaluate the function g(t) = 180(0.8)^t at t = 10.

Answers

1. g(5) = 240(0.75)^5 = 81.5625
2. r = 0.9
3. g(10) = 180(0.8)^10 = 0.1073741824

References

• [1] "Exponential Decay Functions" by Math Is Fun
• [2] "Exponential Functions" by Khan Academy