A Steel Hex Nut Has Two Regular Hexagonal Faces And A Hole With A Diameter Of 0.4 Cm Through The Middle. Its Mass Is 3.03 Grams.Complete The Steps To Calculate The Density Of The Steel:1. Calculate The Volume Of The Prism. Recall That The Area Of A

Understanding the Problem

To calculate the density of a steel hex nut, we need to know its mass and volume. The mass of the steel hex nut is given as 3.03 grams. However, we need to calculate its volume to find its density.

Step 1: Calculate the Volume of the Prism

The steel hex nut has two regular hexagonal faces and a hole with a diameter of 0.4 cm through the middle. To calculate the volume of the prism, we need to find the area of the hexagonal faces and multiply it by the height of the prism.

Area of a Regular Hexagon

A regular hexagon can be divided into six equilateral triangles. To find the area of a regular hexagon, we need to find the area of one equilateral triangle and multiply it by 6.

Area of an Equilateral Triangle

The formula to find the area of an equilateral triangle is:

A = (√3 / 4) × s^2

where s is the side length of the equilateral triangle.

Side Length of the Equilateral Triangle

The side length of the equilateral triangle is equal to the radius of the circle inscribed in the hexagon. Since the diameter of the hole is 0.4 cm, the radius of the circle is half of the diameter, which is 0.2 cm.

Area of the Regular Hexagon

Now that we have the side length of the equilateral triangle, we can find the area of the regular hexagon:

A = 6 × (√3 / 4) × (0.2)^2 A = 6 × (√3 / 4) × 0.04 A = 0.06 × √3 A ≈ 0.1039 cm^2

Volume of the Prism

The volume of the prism is equal to the area of the hexagonal face multiplied by the height of the prism. However, we need to find the height of the prism first.

Height of the Prism

The height of the prism is equal to the distance between the two hexagonal faces. Since the hole is in the middle of the prism, the height of the prism is equal to the diameter of the hole, which is 0.4 cm.

Volume of the Prism

Now that we have the area of the hexagonal face and the height of the prism, we can find the volume of the prism:

V = A × h V = 0.1039 × 0.4 V ≈ 0.04156 cm^3

Step 2: Calculate the Density of the Steel

Now that we have the volume of the prism, we can calculate the density of the steel:

ρ = m / V ρ = 3.03 / 0.04156 ρ ≈ 72.83 g/cm^3

Therefore, the density of the steel hex nut is approximately 72.83 g/cm^3.

Conclusion

Q: What is the formula to calculate the density of a steel hex nut?

A: The formula to calculate the density of a steel hex nut is:

ρ = m / V

where ρ is the density, m is the mass, and V is the volume.

Q: How do I calculate the volume of a steel hex nut?

A: To calculate the volume of a steel hex nut, you need to find the area of the hexagonal face and multiply it by the height of the prism. The area of the hexagonal face can be found by dividing the hexagon into six equilateral triangles and finding the area of one triangle. The height of the prism is equal to the distance between the two hexagonal faces.

Q: What is the formula to find the area of a regular hexagon?

A: The formula to find the area of a regular hexagon is:

A = 6 × (√3 / 4) × s^2

where s is the side length of the equilateral triangle.

Q: How do I find the side length of the equilateral triangle?

A: The side length of the equilateral triangle is equal to the radius of the circle inscribed in the hexagon. Since the diameter of the hole is 0.4 cm, the radius of the circle is half of the diameter, which is 0.2 cm.

Q: What is the formula to find the area of an equilateral triangle?

A: The formula to find the area of an equilateral triangle is:

A = (√3 / 4) × s^2

where s is the side length of the equilateral triangle.

Q: How do I calculate the density of a steel hex nut if I don't know its volume?

A: If you don't know the volume of the steel hex nut, you can calculate it by finding the area of the hexagonal face and multiplying it by the height of the prism. Then, you can use the formula ρ = m / V to calculate the density.

Q: What is the unit of density?

A: The unit of density is typically grams per cubic centimeter (g/cm^3).

Q: Why is it important to calculate the density of a steel hex nut?

A: Calculating the density of a steel hex nut is important because it can help you determine its mass per unit volume, which can be useful in various engineering and scientific applications.

Q: Can I use this method to calculate the density of other objects?

A: Yes, you can use this method to calculate the density of other objects as long as you know their mass and volume. However, you may need to adjust the formula and calculations depending on the shape and size of the object.

Q: What are some common applications of density calculations?

A: Density calculations have many common applications in various fields, including:

• Engineering: to determine the mass per unit volume of materials
• Science: to understand the properties of substances
• Medicine: to calculate the density of tissues and organs
• Industry: to optimize the design and manufacturing of products

Conclusion

In this article, we answered some frequently asked questions about calculating the density of a steel hex nut. We covered topics such as the formula to calculate density, how to calculate the volume of a steel hex nut, and the importance of density calculations. We hope this article has been helpful in understanding the concept of density and how to calculate it.