The Length of the Part on the Drawing

Oct 13, 2025 by miodeya.com 65 views

Understanding Scale Drawings

Scale drawings are used to represent the actual size and proportions of a machine part or an object. They are essential in engineering, architecture, and other fields where precise measurements are crucial. A scale factor is a ratio that compares the size of the drawing to the actual size of the object. In this article, we will explore how to use the scale factor to determine the actual length of a machine part.

The Scale Factor

The scale factor on a scale drawing of a machine part is given as $15^{1 / 8}$ . This means that for every unit of length on the drawing, the actual length of the part is $15^{1 / 8}$ times that unit.

The Length of the Part on the Drawing

The part is $3^{7 / 8}$ inches long on the drawing. To find the actual length of the part, we need to multiply the length on the drawing by the scale factor.

Calculating the Actual Length

Let's calculate the actual length of the part using the given information.

import math
scale_factor = 15 ** (1 / 8)
drawing_length = 3 ** (7 / 8)
actual_length = drawing_length * scale_factor
print("The actual length of the part is:", actual_length)

Simplifying the Calculation

We can simplify the calculation by using the properties of exponents. Since the scale factor is $15^{1 / 8}$ , we can rewrite it as $(3 \cdot 5)^{1 / 8}$ . This allows us to separate the calculation into two parts: one involving the base 3 and the other involving the base 5.

import math
scale_factor = (3 ** (1 / 8)) * (5 ** (1 / 8))
drawing_length = 3 ** (7 / 8)
actual_length = drawing_length * scale_factor
print("The actual length of the part is:", actual_length)

Evaluating the Expression

Now that we have simplified the calculation, we can evaluate the expression to find the actual length of the part.

import math
scale_factor = (3 ** (1 / 8)) * (5 ** (1 / 8))
drawing_length = 3 ** (7 / 8)
actual_length = drawing_length * scale_factor
print("The actual length of the part is:", actual_length)

The Final Answer

After evaluating the expression, we find that the actual length of the part is:

import math
scale_factor = (3 ** (1 / 8)) * (5 ** (1 / 8))
drawing_length = 3 ** (7 / 8)
actual_length = drawing_length * scale_factor
print("The actual length of the part is:", actual_length)

Conclusion

In this article, we have explored how to use the scale factor to determine the actual length of a machine part. We have used the properties of exponents to simplify the calculation and evaluated the expression to find the final answer. The actual length of the part is:

import math
scale_factor = (3 ** (1 / 8)) * (5 ** (1 / 8))
drawing_length = 3 ** (7 / 8)
actual_length = drawing_length * scale_factor
print("The actual length of the part is:", actual_length)

References

Q: What is a scale factor?

A: A scale factor is a ratio that compares the size of a drawing to the actual size of an object. It is used to determine the actual length, width, or height of an object based on its representation on a scale drawing.

Q: How is the scale factor used to determine the actual length of a machine part?

A: The scale factor is used to multiply the length of the part on the drawing by the scale factor to determine the actual length of the part. For example, if the scale factor is $15^{1 / 8}$ and the part is $3^{7 / 8}$ inches long on the drawing, the actual length of the part would be $3^{7 / 8} \cdot 15^{1 / 8}$ inches.

Q: What is the significance of the scale factor in engineering and architecture?

A: The scale factor is crucial in engineering and architecture as it allows designers and engineers to create accurate representations of buildings, bridges, and other structures. It enables them to visualize and plan the project with precision, ensuring that the final product meets the required specifications.

Q: How can the scale factor be used to determine the actual dimensions of a 3D object?

A: The scale factor can be used to determine the actual dimensions of a 3D object by applying it to each dimension separately. For example, if the scale factor is $15^{1 / 8}$ and the object has a length of $3^{7 / 8}$ inches, a width of $2^{5 / 8}$ inches, and a height of $4^{3 / 8}$ inches on the drawing, the actual dimensions of the object would be:

Length: $3^{7 / 8} \cdot 15^{1 / 8}$ inches
Width: $2^{5 / 8} \cdot 15^{1 / 8}$ inches
Height: $4^{3 / 8} \cdot 15^{1 / 8}$ inches

Q: Can the scale factor be used to determine the actual volume of a 3D object?

A: Yes, the scale factor can be used to determine the actual volume of a 3D object by applying it to each dimension separately and then multiplying the results together. For example, if the scale factor is $15^{1 / 8}$ and the object has a length of $3^{7 / 8}$ inches, a width of $2^{5 / 8}$ inches, and a height of $4^{3 / 8}$ inches on the drawing, the actual volume of the object would be:

Actual volume = $(3^{7 / 8} \cdot 15^{1 / 8}) \cdot (2^{5 / 8} \cdot 15^{1 / 8}) \cdot (4^{3 / 8} \cdot 15^{1 / 8})$

Q: What are some common applications of the scale factor in real-world scenarios?

A: The scale factor has numerous applications in real-world scenarios, including:

Architecture: Scale drawings are used to design and plan buildings, bridges, and other structures.
Engineering: Scale drawings are used to design and plan machines, mechanisms, and other devices.
Product design: Scale drawings are used to design and plan products, such as furniture, appliances, and electronics.
Art and design: Scale drawings are used to create accurate representations of artwork and designs.

Q: How can the scale factor be used to determine the actual size of a 2D object?

A: The scale factor can be used to determine the actual size of a 2D object by applying it to each dimension separately. For example, if the scale factor is $15^{1 / 8}$ and the object has a length of $3^{7 / 8}$ inches and a width of $2^{5 / 8}$ inches on the drawing, the actual size of the object would be:

Actual length: $3^{7 / 8} \cdot 15^{1 / 8}$ inches
Actual width: $2^{5 / 8} \cdot 15^{1 / 8}$ inches

Q: Can the scale factor be used to determine the actual area of a 2D object?

A: Yes, the scale factor can be used to determine the actual area of a 2D object by applying it to each dimension separately and then multiplying the results together. For example, if the scale factor is $15^{1 / 8}$ and the object has a length of $3^{7 / 8}$ inches and a width of $2^{5 / 8}$ inches on the drawing, the actual area of the object would be:

Actual area = $(3^{7 / 8} \cdot 15^{1 / 8}) \cdot (2^{5 / 8} \cdot 15^{1 / 8})$

Conclusion

In this Q&A article, we have explored the concept of the scale factor and its applications in determining the actual size and dimensions of objects. We have also discussed how the scale factor can be used to determine the actual volume and area of 3D and 2D objects, respectively. The scale factor is a crucial tool in engineering, architecture, and product design, and its applications are numerous and varied.