If the right triangular prism is dilated using a scale factor of 3, what would be the volume of the dilated prism? (1 point)

Question

sides are 3 cm, 4 cm, 7 cm
make sure to dilate

Answer 1

To find the volume of a right triangular prism after dilation, we first need to calculate the original volume of the prism and then use the scale factor.

Step 1: Calculate the Volume of the Original Prism

The formula for the volume \( V \) of a prism is given by:

\[ V = B \times h \]

where \( B \) is the area of the base and \( h \) is the height of the prism.

Step 1.1: Find the Area of the Triangular Base

The area \( A \) of a right triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

In this case, we can take the sides of the triangle as 3 cm and 4 cm as the base and height (the right-angled sides). Thus,

\[ A = \frac{1}{2} \times 3 \times 4 = \frac{12}{2} = 6 \text{ cm}^2 \]

Step 1.2: Determine the Height of the Prism

The height of the prism is given as 7 cm.

Step 1.3: Calculate the Volume

Plugging the area of the base and the height back into the volume formula:

\[ V = A \times h = 6 \text{ cm}^2 \times 7 \text{ cm} = 42 \text{ cm}^3 \]

Step 2: Dilate the Prism

When a prism is dilated using a scale factor \( k \), the volume of the dilated prism changes according to the cube of the scale factor:

\[ V_{\text{dilated}} = k^3 \times V_{\text{original}} \]

In this case, the scale factor \( k = 3 \):

\[ V_{\text{dilated}} = 3^3 \times 42 \text{ cm}^3 = 27 \times 42 \text{ cm}^3 \]

Calculating that:

\[ 27 \times 42 = 1134 \text{ cm}^3 \]

Conclusion

The volume of the dilated prism is

\[ \boxed{1134 \text{ cm}^3} \]

Answer 2

that's not dilating that's making it bigger

Answer 3

When we discuss the concept of "dilating" a geometric figure, it refers to the process of enlarging or reducing it by a specific scale factor.

In the previous message, I interpreted your request correctly, but let’s clarify how the term "dilate" is used in geometry. The word itself implies an enlargement when the scale factor is greater than 1. So when a right triangular prism's dimensions are scaled by a factor of 3, you are indeed making it larger.

Understanding Dilation

Here's how dilation affects the volume of the prism:

1. Original Volume: You start with a certain volume.
2. Scale Factor: When you apply a dilation with a scale factor \( k \) (in this case, \( k = 3 \)), all dimensions (length, width, height) of the prism are multiplied by this factor.
3. New Volume Calculation: Volume scales with the cube of the scale factor, hence the new volume is calculated as:

\[ V_{\text{dilated}} = k^3 \times V_{\text{original}} \]

Recap of Volume Calculation

1. Original Dimensions:

   • Base: \(3 \text{ cm}, 4 \text{ cm}\) (height of the triangle)
   • Height of the prism: \(7 \text{ cm}\)

2. Original Volume: \[ V_{\text{original}} = A \times h = 6 \text{ cm}^2 \times 7 \text{ cm} = 42 \text{ cm}^3 \]

3. Applying Dilation:

   • New dimensions after dilation will be:
     • Base (3 cm) becomes \(3 \times 3 = 9 \text{ cm}\)
     • Height (4 cm) becomes \(4 \times 3 = 12 \text{ cm}\)
     • Height of the prism (7 cm) becomes \(7 \times 3 = 21 \text{ cm}\)

4. New Area of the Base: \[ A_{\text{dilated}} = \frac{1}{2} \times 9 \times 12 = \frac{108}{2} = 54 \text{ cm}^2 \]

5. New Volume: \[ V_{\text{dilated}} = A_{\text{dilated}} \times h_{\text{dilated}} = 54 \text{ cm}^2 \times 21 \text{ cm} = 1134 \text{ cm}^3 \]

Summary

The operation was indeed about increasing the size (dilating) of the prism. The calculated volume of the dilated prism, after multiplying the dimensions of the original prism by a scale factor of 3, is

\[ \boxed{1134 \text{ cm}^3} \]

If you're referring to something different by "dilating" or meant to apply a reduction or different type of transformation, please clarify, and I’d be happy to assist further!