Type The Correct Answer In The Box.The Value Of An Autographed Baseball From 2017 Is $300$. The Value Of The Baseball Exponentially Increases By 5 % 5 \% 5% Each Year. Write A One-variable Inequality That Could Be Used To Solve For The

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Understanding the Problem

In this problem, we are given the value of an autographed baseball from 2017, which is $300. The value of the baseball is said to increase exponentially by 5% each year. We need to write a one-variable inequality that could be used to solve for the value of the baseball after a certain number of years.

The Exponential Growth Formula

The exponential growth formula is given by:

A = P(1 + r)^t

Where:

  • A is the final amount
  • P is the initial amount
  • r is the rate of growth (in decimal form)
  • t is the time period (in years)

In this case, the initial amount (P) is $300, the rate of growth (r) is 5% or 0.05, and we want to find the final amount (A) after a certain number of years (t).

Writing the Inequality

Since the value of the baseball increases exponentially, we can write an inequality to represent this situation. Let's say the value of the baseball after t years is greater than or equal to $300. We can write this as:

300(1 + 0.05)^t ≥ A

Simplifying the Inequality

We can simplify the inequality by substituting the values of P and r:

300(1.05)^t ≥ A

The One-Variable Inequality

Now, we can write the one-variable inequality that could be used to solve for the value of the baseball after a certain number of years:

300(1.05)^t ≥ A

Solving the Inequality

To solve the inequality, we need to isolate the variable t. We can do this by dividing both sides of the inequality by 300:

(1.05)^t ≥ A/300

Taking the Logarithm

To solve for t, we can take the logarithm of both sides of the inequality. We can use any base for the logarithm, but let's use the natural logarithm (ln):

ln((1.05)^t) ≥ ln(A/300)

Using the Power Rule

Using the power rule of logarithms, we can rewrite the left-hand side of the inequality:

t ln(1.05) ≥ ln(A/300)

Solving for t

Now, we can solve for t by dividing both sides of the inequality by ln(1.05):

t ≥ ln(A/300) / ln(1.05)

The Final Answer

The one-variable inequality that could be used to solve for the value of the baseball after a certain number of years is:

t ≥ ln(A/300) / ln(1.05)

This inequality can be used to find the value of t for any given value of A.

Conclusion

In this problem, we were given the value of an autographed baseball from 2017, which is $300. The value of the baseball is said to increase exponentially by 5% each year. We wrote a one-variable inequality that could be used to solve for the value of the baseball after a certain number of years. The inequality is:

t ≥ ln(A/300) / ln(1.05)

This inequality can be used to find the value of t for any given value of A.

Example Use Case

Let's say we want to find the value of t when the value of the baseball is $500. We can plug in the value of A into the inequality:

t ≥ ln(500/300) / ln(1.05)

Simplifying the inequality, we get:

t ≥ 2.01

So, the value of t when the value of the baseball is $500 is approximately 2.01 years.

Real-World Applications

The exponential growth formula and the one-variable inequality can be used to model many real-world situations, such as:

  • Population growth
  • Financial investments
  • Chemical reactions
  • Biological systems

In each of these situations, the exponential growth formula can be used to model the growth or decay of a quantity over time. The one-variable inequality can then be used to solve for the time period required to reach a certain value.

Limitations

The exponential growth formula and the one-variable inequality have some limitations. For example:

  • The formula assumes that the rate of growth is constant over time.
  • The formula assumes that the initial amount is known.
  • The inequality assumes that the value of the baseball is greater than or equal to $300.

In some cases, these assumptions may not be valid, and the formula and inequality may not accurately model the situation.

Future Research

Future research could focus on developing more accurate models of exponential growth and decay. This could involve:

  • Developing more complex models that take into account multiple factors
  • Using data from real-world situations to test and validate the models
  • Developing new methods for solving the one-variable inequality

Q: What is exponential growth?

A: Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the value of the baseball increases by a fixed percentage each year, rather than a fixed amount.

Q: How does the exponential growth formula work?

A: The exponential growth formula is given by:

A = P(1 + r)^t

Where:

  • A is the final amount
  • P is the initial amount
  • r is the rate of growth (in decimal form)
  • t is the time period (in years)

Q: What is the rate of growth (r) in the problem?

A: The rate of growth (r) is 5% or 0.05.

Q: How do I use the one-variable inequality to solve for t?

A: To solve for t, you can use the one-variable inequality:

t ≥ ln(A/300) / ln(1.05)

Q: What is the value of t when the value of the baseball is $500?

A: To find the value of t when the value of the baseball is $500, you can plug in the value of A into the inequality:

t ≥ ln(500/300) / ln(1.05)

Simplifying the inequality, you get:

t ≥ 2.01

So, the value of t when the value of the baseball is $500 is approximately 2.01 years.

Q: Can I use the exponential growth formula and the one-variable inequality to model other real-world situations?

A: Yes, the exponential growth formula and the one-variable inequality can be used to model many real-world situations, such as:

  • Population growth
  • Financial investments
  • Chemical reactions
  • Biological systems

Q: What are some limitations of the exponential growth formula and the one-variable inequality?

A: Some limitations of the exponential growth formula and the one-variable inequality include:

  • The formula assumes that the rate of growth is constant over time.
  • The formula assumes that the initial amount is known.
  • The inequality assumes that the value of the baseball is greater than or equal to $300.

Q: How can I develop more accurate models of exponential growth and decay?

A: To develop more accurate models of exponential growth and decay, you can:

  • Develop more complex models that take into account multiple factors
  • Use data from real-world situations to test and validate the models
  • Develop new methods for solving the one-variable inequality

Q: What are some real-world applications of the exponential growth formula and the one-variable inequality?

A: Some real-world applications of the exponential growth formula and the one-variable inequality include:

  • Predicting population growth and decline
  • Modeling financial investments and returns
  • Understanding chemical reactions and biological systems
  • Making informed decisions in a wide range of fields

Q: Can I use the exponential growth formula and the one-variable inequality to solve problems in other areas of mathematics?

A: Yes, the exponential growth formula and the one-variable inequality can be used to solve problems in other areas of mathematics, such as:

  • Calculus
  • Algebra
  • Geometry
  • Statistics

Q: How can I use the exponential growth formula and the one-variable inequality to make informed decisions in real-world situations?

A: To use the exponential growth formula and the one-variable inequality to make informed decisions in real-world situations, you can:

  • Use the formula and inequality to predict and model growth and decay
  • Use data from real-world situations to test and validate the models
  • Develop new methods for solving the one-variable inequality
  • Make informed decisions based on the results of the models and inequality.