Question

A triangular prism has a base area of 28 cm2 and a height of 13 cm. If the prism is dilated with a scale factor of k = 0.5, what is the new volume?

a. 45.5 cm3
b. 91 cm3
c. 182 cm3
d. 364 cm3

Answer 1

The new volume of the dilated prism is 11.375 cm³.

Step-by-step explanation:

To find the new volume of a dilated triangular prism, we need to apply the scale factor to the base area and height. The scale factor is 0.5, so we multiply the base area of 28 cm² by 0.5² and the height of 13 cm by 0.5. Then, we use the formula for the volume of a triangular prism: Volume = Base Area × Height. Plugging in the values, we get:

Volume = (28 cm² × 0.5²) × (13 cm × 0.5)

Volume = 7 cm² × 0.25 × 6.5 cm

Volume = 11.375 cm³

Therefore, the new volume of the dilated prism is 11.375 cm³.

Answer 2

the new volume of the prism after the dilation is 45.5 cm³, which corresponds to option A.

To find the new volume of a dilated triangular prism, we'll need to consider how the dilation scale factor affects the volume of a three-dimensional object.

The original volume
`\( V \)` of a triangular prism is given by the formula:

`\[ V = \text{base area} * \text{height} \]`

The new volume of the prism after the dilation is 45.5 cm³, which corresponds to option A.

For the original prism:

`\[ V_{\text{original}} = 28 \, \text{cm}^2 * 13 \, \text{cm} = 364 \, \text{cm}^3 \]`

When the prism is dilated by a scale factor \( k \), each linear dimension of the prism is multiplied by \( k \). The volume, being a three-dimensional measure, is affected by the cube of the scale factor, because the volume is a function of three linear dimensions (length, width, and height).

`\[ V_{\text{new}} = k^3 * V_{\text{original}} \]`

Given that
`\( k = 0.5 \)`, the new volume is:

`\[ V_{\text{new}} = (0.5)^3 * 364 \, \text{cm}^3 = (1)/(8) * 364 \, \text{cm}^3 \]`

Now let's calculate the new volume step-wise:

`\[ V_{\text{new}} = (1)/(8) * 364 \]`

`\[ V_{\text{new}} = (364)/(8) \]`

`\[ V_{\text{new}} = 45.5 \, \text{cm}^3 \]`

Therefore, the new volume of the prism after the dilation is 45.5 cm³, which corresponds to option A.