Question

A square prism has base side lengths measuring 4 yards and a height of 11 yards. if the square prism is dilated by a scale factor of k = 2, what is the new volume?

Answer 1

The new volume of a square prism dilated by a scale factor of 2, with original dimensions of 4 yards by 4 yards by 11 yards, is 1408 cubic yards. We find this by cubing the scale factor and multiplying it by the original volume.

Step-by-step explanation:

The student is asking about a geometric transformation known as dilation and its effect on the volume of a square prism. To find the new volume after dilation, we calculate the volume of the original prism and then apply the scale factor. For the original prism with base side lengths of 4 yards and a height of 11 yards, the volume is calculated as:

• Volume of the original prism = base area × height = (4 yards × 4 yards) × 11 yards = 16 yards² × 11 yards = 176 yards³.

When the square prism is dilated by a scale factor of k = 2, the dimensions of the prism are doubled. Therefore, the new base side length is 8 yards (as 4 yards × 2). The new volume can then be calculated using the fact that for dilation in three dimensions, the change in volume is the cube of the scale factor, so the new volume is:

• New volume = (scale factor)³ × original volume = 2³ × 176 yards³ = 8 × 176 yards³ = 1408 yards³.

Thus, the new volume of the dilated square prism is 1408 cubic yards.

Answer 2

The new volume of the dilated square prism is 1,408 cubic yards.

The volume of a square prism can be calculated by multiplying the area of the base by the height.

In this case, the base of the square prism has side lengths measuring 4 yards, so the area of the base is 4 yards multiplied by 4 yards, which equals 16 square yards.

The height of the square prism is given as 11 yards.

To find the new volume after dilating the square prism by a scale factor of k = 2, we need to multiply the original volume by the cube of the scale factor.

The original volume is calculated by multiplying the base area (16 square yards) by the height (11 yards), resulting in a volume of 176 cubic yards.

To find the new volume, we multiply the original volume by the cube of the scale factor. Since the scale factor is 2, we cube it: 2^3 = 8.

Multiplying the original volume (176 cubic yards) by the cube of the scale factor (8) gives us the new volume.

176 cubic yards multiplied by 8 equals 1,408 cubic yards.