Question

a cube has edge length 3 inches.
a. find the surface area and volume of the cube
surface area = square inches
volume = cubic inches

b. the cube is dilated by a scale factor of 0.5. find the surface area and volume of the image
write your answers as decimals.
surface area = square inches
volume = cubic inches

Explanation:

Part a

Step 1: Surface Area of Cube

The formula for the surface area (\(SA\)) of a cube is \(SA = 6s^2\), where \(s\) is the edge length. Here, \(s = 3\) inches.
\[
SA = 6\times(3)^2 = 6\times9 = 54
\]

Step 2: Volume of Cube

The formula for the volume (\(V\)) of a cube is \(V = s^3\). For \(s = 3\) inches:
\[
V = (3)^3 = 27
\]

Step 1: New Edge Length after Dilation

The scale factor is \(0.5\), so the new edge length (\(s'\)) is \(s\times0.5 = 3\times0.5 = 1.5\) inches.

Step 2: Surface Area of Dilated Cube

Using the surface area formula \(SA' = 6(s')^2\) with \(s' = 1.5\):
\[
SA' = 6\times(1.5)^2 = 6\times2.25 = 13.5
\]

Step 3: Volume of Dilated Cube

Using the volume formula \(V' = (s')^3\) with \(s' = 1.5\):
\[
V' = (1.5)^3 = 3.375
\]

Answer:

Surface Area = \(54\) square inches
Volume = \(27\) cubic inches

Part b