Combining the Inequalities

Oct 13, 2025 by wbrum.org 93 views

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Problem Description

A sugar solution currently measures 250 degrees Fahrenheit. The temperature of the solution increases 5 degrees per minute. The sugar solution will caramelize at temperatures greater than 310 and less than 339 degrees Fahrenheit. Which inequality can be used to represent the temperature of the sugar solution as a function of time?

Understanding the Problem

To solve this problem, we need to understand the relationship between the temperature of the sugar solution and time. The temperature of the solution increases by 5 degrees per minute, and we need to find an inequality that represents the temperature of the solution as a function of time.

Setting Up the Inequality

Let's denote the temperature of the sugar solution as T (in degrees Fahrenheit) and the time as t (in minutes). Since the temperature increases by 5 degrees per minute, we can write the equation:

T = 250 + 5t

We want to find an inequality that represents the temperature of the solution as a function of time, such that the solution caramelize at temperatures greater than 310 and less than 339 degrees Fahrenheit.

Writing the Inequality

To write the inequality, we need to consider the two conditions:

The temperature of the solution is greater than 310 degrees Fahrenheit.
The temperature of the solution is less than 339 degrees Fahrenheit.

We can write these conditions as inequalities:

T > 310 T < 339

Combining the Inequalities

Since we want to find an inequality that represents the temperature of the solution as a function of time, we can combine the two inequalities:

310 < T < 339

Substituting the Equation

We can substitute the equation T = 250 + 5t into the inequality:

310 < 250 + 5t < 339

Simplifying the Inequality

To simplify the inequality, we can subtract 250 from all three parts:

60 < 5t < 89

Dividing by 5

To isolate t, we can divide all three parts by 5:

12 < t < 17.8

Conclusion

The inequality that represents the temperature of the sugar solution as a function of time is:

12 < t < 17.8

This means that the sugar solution will caramelize between 12 and 17.8 minutes after the initial temperature of 250 degrees Fahrenheit.

Final Answer

The final answer is:

12 < t < 17.8

This inequality represents the temperature of the sugar solution as a function of time, such that the solution caramelize at temperatures greater than 310 and less than 339 degrees Fahrenheit.

Example Use Case

This problem can be used to model real-world scenarios, such as cooking or food processing. For example, a chef may want to know how long it will take for a sugar solution to caramelize at a certain temperature, or a food processor may want to know how long it will take for a sugar solution to reach a certain temperature.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Write the equation T = 250 + 5t to represent the temperature of the sugar solution as a function of time.
Write the inequalities T > 310 and T < 339 to represent the conditions for caramelization.
Combine the inequalities to get 310 < T < 339.
Substitute the equation T = 250 + 5t into the inequality.
Simplify the inequality to get 60 < 5t < 89.
Divide all three parts by 5 to get 12 < t < 17.8.

Key Concepts

Inequalities: Inequalities are used to represent relationships between variables that are not equal.
Functions: Functions are used to represent relationships between variables.
Temperature: Temperature is a measure of the heat energy of a substance.
Caramelization: Caramelization is the process of converting sugar into a brown, sweet, and sticky substance.

References

[1] "Inequalities and Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f1f1f1/x2f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f1f

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Q: What is the initial temperature of the sugar solution?

A: The initial temperature of the sugar solution is 250 degrees Fahrenheit.

Q: How fast is the temperature of the sugar solution increasing?

A: The temperature of the sugar solution is increasing by 5 degrees per minute.

Q: What are the conditions for caramelization?

A: The sugar solution will caramelize at temperatures greater than 310 and less than 339 degrees Fahrenheit.

Q: What is the inequality that represents the temperature of the sugar solution as a function of time?

A: The inequality that represents the temperature of the sugar solution as a function of time is:

12 < t < 17.8

Q: What does this inequality mean?

A: This inequality means that the sugar solution will caramelize between 12 and 17.8 minutes after the initial temperature of 250 degrees Fahrenheit.

Q: How can this problem be used in real-world scenarios?

A: This problem can be used to model real-world scenarios, such as cooking or food processing. For example, a chef may want to know how long it will take for a sugar solution to caramelize at a certain temperature, or a food processor may want to know how long it will take for a sugar solution to reach a certain temperature.

Q: What are some key concepts related to this problem?

A: Some key concepts related to this problem include:

Inequalities: Inequalities are used to represent relationships between variables that are not equal.
Functions: Functions are used to represent relationships between variables.
Temperature: Temperature is a measure of the heat energy of a substance.
Caramelization: Caramelization is the process of converting sugar into a brown, sweet, and sticky substance.

Q: How can this problem be solved step-by-step?

A: Here is a step-by-step solution to the problem:

Write the equation T = 250 + 5t to represent the temperature of the sugar solution as a function of time.
Write the inequalities T > 310 and T < 339 to represent the conditions for caramelization.
Combine the inequalities to get 310 < T < 339.
Substitute the equation T = 250 + 5t into the inequality.
Simplify the inequality to get 60 < 5t < 89.
Divide all three parts by 5 to get 12 < t < 17.8.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

Not writing the equation T = 250 + 5t to represent the temperature of the sugar solution as a function of time.
Not writing the inequalities T > 310 and T < 339 to represent the conditions for caramelization.
Not combining the inequalities to get 310 < T < 339.
Not substituting the equation T = 250 + 5t into the inequality.
Not simplifying the inequality to get 60 < 5t < 89.
Not dividing all three parts by 5 to get 12 < t < 17.8.

Q: How can this problem be used to model real-world scenarios?

Q: What are some applications of this problem in real-world scenarios?

A: Some applications of this problem in real-world scenarios include:

Cooking: A chef may want to know how long it will take for a sugar solution to caramelize at a certain temperature.
Food processing: A food processor may want to know how long it will take for a sugar solution to reach a certain temperature.
Chemical engineering: A chemical engineer may want to know how long it will take for a sugar solution to caramelize at a certain temperature.

Q: How can this problem be used to model complex systems?

A: This problem can be used to model complex systems, such as cooking or food processing. For example, a chef may want to know how long it will take for a sugar solution to caramelize at a certain temperature, or a food processor may want to know how long it will take for a sugar solution to reach a certain temperature.

Q: What are some limitations of this problem?

A: Some limitations of this problem include:

The problem assumes that the temperature of the sugar solution is increasing at a constant rate.
The problem assumes that the sugar solution will caramelize at temperatures greater than 310 and less than 339 degrees Fahrenheit.
The problem does not take into account other factors that may affect the caramelization of the sugar solution, such as the presence of other substances or the temperature of the surrounding environment.