Select The Correct Answer.A Triangle Has Side Lengths Of $n$, $n-3$, And $2(n-2$\]. If The Perimeter Of The Triangle Is At Least 37 Units, What Is The Value Of $n$?A. $n \geq 11$ B. $n \geq 7.5$ C.

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Understanding the Problem

In this problem, we are given a triangle with side lengths of nn, nāˆ’3n-3, and 2(nāˆ’2)2(n-2). The perimeter of the triangle is at least 37 units, and we need to find the value of nn that satisfies this condition.

Defining the Perimeter

The perimeter of a triangle is the sum of the lengths of its three sides. In this case, the perimeter is given by the expression:

P=n+(nāˆ’3)+2(nāˆ’2)P = n + (n-3) + 2(n-2)

Simplifying the Perimeter Expression

To simplify the perimeter expression, we can combine like terms:

P=n+nāˆ’3+2nāˆ’4P = n + n - 3 + 2n - 4

P=4nāˆ’7P = 4n - 7

Setting Up the Inequality

Since the perimeter is at least 37 units, we can set up the following inequality:

4nāˆ’7ā‰„374n - 7 \geq 37

Solving the Inequality

To solve the inequality, we can add 7 to both sides:

4nā‰„444n \geq 44

Next, we can divide both sides by 4:

nā‰„11n \geq 11

Evaluating the Answer Choices

Now that we have found the value of nn, we can evaluate the answer choices:

  • A. nā‰„11n \geq 11: This is the correct answer, as we found that nā‰„11n \geq 11.
  • B. nā‰„7.5n \geq 7.5: This is not the correct answer, as we found that nā‰„11n \geq 11, which is greater than 7.5.
  • C. nā‰„12n \geq 12: This is not the correct answer, as we found that nā‰„11n \geq 11, which is less than 12.

Conclusion

In this problem, we were given a triangle with side lengths of nn, nāˆ’3n-3, and 2(nāˆ’2)2(n-2). We found that the perimeter of the triangle is at least 37 units, and we solved for the value of nn that satisfies this condition. The correct answer is nā‰„11n \geq 11.

Additional Considerations

It's worth noting that the value of nn must be a positive number, as the side lengths of a triangle cannot be negative. Additionally, the value of nn must be an integer, as the side lengths of a triangle must be whole numbers.

Real-World Applications

This problem has real-world applications in various fields, such as engineering and architecture. For example, when designing a building or a bridge, engineers need to consider the perimeter of the structure to ensure that it is stable and secure.

Final Thoughts

In conclusion, this problem requires the application of mathematical concepts, such as inequalities and algebraic expressions, to solve for the value of nn. The correct answer is nā‰„11n \geq 11, and this problem has real-world applications in various fields.

Step-by-Step Solution

  1. Define the perimeter of the triangle: P=n+(nāˆ’3)+2(nāˆ’2)P = n + (n-3) + 2(n-2)
  2. Simplify the perimeter expression: P=4nāˆ’7P = 4n - 7
  3. Set up the inequality: 4nāˆ’7ā‰„374n - 7 \geq 37
  4. Solve the inequality: nā‰„11n \geq 11

Key Concepts

  • Perimeter of a triangle
  • Inequalities
  • Algebraic expressions
  • Real-world applications of mathematics

Common Mistakes

  • Failing to simplify the perimeter expression
  • Failing to set up the inequality correctly
  • Failing to solve the inequality correctly

Tips and Tricks

  • Make sure to simplify the perimeter expression before setting up the inequality.
  • Use algebraic properties to solve the inequality.
  • Check the answer choices to ensure that they are correct.
    Frequently Asked Questions (FAQs) =====================================

Q: What is the perimeter of a triangle?

A: The perimeter of a triangle is the sum of the lengths of its three sides.

Q: How do I find the perimeter of a triangle?

A: To find the perimeter of a triangle, you need to add the lengths of all three sides together. For example, if a triangle has side lengths of 3, 4, and 5, the perimeter would be 3 + 4 + 5 = 12.

Q: What is the formula for the perimeter of a triangle?

A: The formula for the perimeter of a triangle is P = a + b + c, where P is the perimeter and a, b, and c are the lengths of the three sides.

Q: How do I solve an inequality like 4n - 7 ā‰„ 37?

A: To solve an inequality like 4n - 7 ā‰„ 37, you need to add 7 to both sides of the inequality, which gives you 4n ā‰„ 44. Then, you can divide both sides by 4, which gives you n ā‰„ 11.

Q: What is the value of n in the problem?

A: The value of n in the problem is n ā‰„ 11.

Q: Why is the value of n important?

A: The value of n is important because it determines the minimum length of the sides of the triangle. If n is too small, the triangle may not be valid.

Q: What are some real-world applications of the perimeter of a triangle?

A: Some real-world applications of the perimeter of a triangle include:

  • Building design: Architects use the perimeter of a building to determine the amount of materials needed for construction.
  • Bridge design: Engineers use the perimeter of a bridge to determine the amount of materials needed for construction.
  • Landscaping: Gardeners use the perimeter of a garden to determine the amount of materials needed for landscaping.

Q: How do I apply the concept of perimeter to real-world problems?

A: To apply the concept of perimeter to real-world problems, you need to identify the lengths of the sides of the object or structure and add them together to find the perimeter. Then, you can use the perimeter to determine the amount of materials needed for construction or other purposes.

Q: What are some common mistakes to avoid when working with the perimeter of a triangle?

A: Some common mistakes to avoid when working with the perimeter of a triangle include:

  • Failing to simplify the perimeter expression
  • Failing to set up the inequality correctly
  • Failing to solve the inequality correctly

Q: How do I check my work when solving a problem involving the perimeter of a triangle?

A: To check your work when solving a problem involving the perimeter of a triangle, you need to:

  • Verify that the perimeter expression is simplified correctly
  • Verify that the inequality is set up correctly
  • Verify that the inequality is solved correctly

Additional Resources

  • [Mathematics textbooks]
  • [Online math resources]
  • [Mathematical software]

Conclusion

In conclusion, the perimeter of a triangle is an important concept in mathematics that has real-world applications in various fields. By understanding how to find the perimeter of a triangle and how to solve inequalities, you can apply this concept to real-world problems and make informed decisions.

Key Takeaways

  • The perimeter of a triangle is the sum of the lengths of its three sides.
  • The formula for the perimeter of a triangle is P = a + b + c.
  • To solve an inequality like 4n - 7 ā‰„ 37, you need to add 7 to both sides and then divide both sides by 4.
  • The value of n in the problem is n ā‰„ 11.
  • The perimeter of a triangle has real-world applications in various fields, including building design, bridge design, and landscaping.

Common Misconceptions

  • The perimeter of a triangle is only used in mathematics.
  • The perimeter of a triangle is only used in geometry.
  • The perimeter of a triangle is only used in real-world problems involving shapes.

Tips and Tricks

  • Make sure to simplify the perimeter expression before setting up the inequality.
  • Use algebraic properties to solve the inequality.
  • Check the answer choices to ensure that they are correct.