Determine the volume of the rectangular prism if it were dilated using scale factor of 4. If the answer is 1,536 m^3 can you please explain how it is the answer

5 answers

To determine the volume of a rectangular prism after dilation, you need to know the original dimensions and how dilation affects volume.

  1. Understanding Dilation: When you dilate a three-dimensional shape by a scale factor \( k \), each dimension (length, width, height) of the shape is multiplied by \( k \). Therefore, the formula for the new volume \( V' \) after dilation is: \[ V' = k^3 \times V \] where \( V \) is the original volume of the prism.

  2. Given Scale Factor: You mentioned a scale factor of 4. Therefore, \( k = 4 \), and we need to calculate \( k^3 \): \[ k^3 = 4^3 = 64 \]

  3. Original Volume: Next, you would need the original volume \( V \) of the prism. If the volume after dilation is given as \( 1,536 , \text{m}^3 \), we can set up the equation: \[ V' = 64 \times V \] So, \[ 1,536 = 64 \times V \]

  4. Solve for \( V \): To find the original volume \( V \): \[ V = \frac{1,536}{64} \] Calculating this gives: \[ V = 24 , \text{m}^3 \]

  5. Conclusion: Therefore, if the original volume of the rectangular prism is \( 24 , \text{m}^3 \), and when dilated by a scale factor of 4, the volume becomes \( 1,536 , \text{m}^3 \) as you mentioned.

In summary, if the volume of the rectangular prism after dilation is \( 1,536 , \text{m}^3 \), it indicates that the original volume was \( 24 , \text{m}^3 \) before being dilated by a scale factor of \( 4 \).

Oh I'm sorry the dimensions are 4 m by 3 m by 2 m

No problem! Let's calculate the volume of the rectangular prism with the given dimensions \(4 , \text{m}\), \(3 , \text{m}\), and \(2 , \text{m}\), and see what happens when we apply a dilation with a scale factor of \(4\).

Step 1: Calculate the Original Volume

The volume \(V\) of a rectangular prism is calculated using the formula: \[ V = \text{length} \times \text{width} \times \text{height} \]

Given the dimensions:

  • Length = \(4 , \text{m}\)
  • Width = \(3 , \text{m}\)
  • Height = \(2 , \text{m}\)

Now plug in the values: \[ V = 4 \times 3 \times 2 \]

Calculating this: \[ V = 12 \times 2 = 24 , \text{m}^3 \]

Step 2: Calculate the Volume After Dilation

Now, when we dilate the rectangular prism by a scale factor of \(4\), we will multiply each dimension by \(4\):

  • New Length = \(4 \times 4 = 16 , \text{m}\)
  • New Width = \(3 \times 4 = 12 , \text{m}\)
  • New Height = \(2 \times 4 = 8 , \text{m}\)

Now, calculate the new volume \(V'\): \[ V' = \text{New Length} \times \text{New Width} \times \text{New Height} \] \[ V' = 16 \times 12 \times 8 \]

Calculating this step by step:

  1. \(16 \times 12 = 192\)
  2. \(192 \times 8 = 1536\)

So the volume after dilation is: \[ V' = 1536 , \text{m}^3 \]

Conclusion

Thus, the volume of the rectangular prism after applying the dilation with a scale factor of \(4\) is indeed \(1536 , \text{m}^3\). Thank you for providing the dimensions!

Thank you I understand now

You're welcome! I'm glad to hear that you understand now. If you have any more questions about geometry, volume, or any other topic, feel free to ask. Happy studying!