Determine the volume of the figure if it were dilated using a scale factor of 2. (1 point) cm3

Question

height is 6
width is 8
length is 10

Answer 1

To determine the volume of the given figure before and after dilation, we can follow these steps:

1. Calculate the original volume of the figure: The volume \( V \) of a rectangular prism (or box) can be calculated using the formula: \[ V = \text{length} \times \text{width} \times \text{height} \] Given the dimensions:

   • Length = 10 cm
   • Width = 8 cm
   • Height = 6 cm

   Plugging in the values: \[ V = 10 , \text{cm} \times 8 , \text{cm} \times 6 , \text{cm} = 480 , \text{cm}^3 \]

2. Calculate the volume after dilation: When an object is dilated by a scale factor \( k \), its volume is multiplied by \( k^3 \). In this case, the scale factor \( k \) is 2. \[ \text{New Volume} = V \times k^3 = V \times 2^3 \] Since \( 2^3 = 8 \): \[ \text{New Volume} = 480 , \text{cm}^3 \times 8 = 3840 , \text{cm}^3 \]

Thus, the volume of the figure after dilation with a scale factor of 2 is 3840 cm³.

Answer 2

it says its wrong

Answer 3

Let’s double-check and clarify the calculation step-by-step.

1. Calculate the original volume of the rectangular prism:

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

   • Length = 10 cm
   • Width = 8 cm
   • Height = 6 cm

   Now, substituting these values into the formula: \[ V = 10 , \text{cm} \times 8 , \text{cm} \times 6 , \text{cm} \] \[ V = 10 \times 8 = 80 \] \[ V = 80 \times 6 = 480 , \text{cm}^3 \]

   The original volume is indeed 480 cm³.

2. Calculate the volume after dilation with a scale factor of 2:

   When an object is dilated by a scale factor \( k \), the volume is scaled by \( k^3 \).

   Here the scale factor \( k \) is 2: \[ k^3 = 2^3 = 8 \]

   Therefore, the new volume after dilation becomes: \[ \text{New Volume} = V \times k^3 = 480 , \text{cm}^3 \times 8 \] Calculating this: \[ \text{New Volume} = 480 \times 8 = 3840 , \text{cm}^3 \]

So the calculation confirms that the volume of the figure after dilation with a scale factor of 2 is indeed 3840 cm³. If it still indicates a mistake, could you please provide additional context about any specific requirements or conditions that might apply?

Answer 4

it says dilated, I think you're expanding

Answer 5

Let's clarify the process of dilation and confirm the calculations regarding the volume of the figure after dilation.

When a three-dimensional shape is dilated using a scale factor \( k \), its dimensions are scaled by \( k \), and the volumes are affected by the cube of the scale factor.

1. Determine the Original Volume:

   We already calculated the original volume of the rectangular prism: \[ V = \text{length} \times \text{width} \times \text{height} = 10 , \text{cm} \times 8 , \text{cm} \times 6 , \text{cm} = 480 , \text{cm}^3. \]

2. Applying the Scale Factor to the Dimensions:

   To find the new dimensions after dilation with a scale factor of 2:

   • New length = \( 10 , \text{cm} \times 2 = 20 , \text{cm} \)
   • New width = \( 8 , \text{cm} \times 2 = 16 , \text{cm} \)
   • New height = \( 6 , \text{cm} \times 2 = 12 , \text{cm} \)

3. Calculate the New Volume Using New Dimensions:

   Now calculate the new volume with the new dimensions: \[ V_{\text{new}} = \text{new length} \times \text{new width} \times \text{new height} \] \[ V_{\text{new}} = 20 , \text{cm} \times 16 , \text{cm} \times 12 , \text{cm} \]

   First, calculate \( 20 \times 16 \): \[ 20 \times 16 = 320 , \text{cm}^2 \]

   Then multiply by the height: \[ 320 , \text{cm}^2 \times 12 , \text{cm} = 3840 , \text{cm}^3 \]

Thus, the resulting volume after dilation is indeed 3840 cm³.

If you are looking for a different interpretation of "dilated" in terms of only applying the scale factor to the original volume without recalculating the dimensions, please clarify that context. However, according to standard geometric principles, the volume increases as calculated above.