Question

consider the cube shaped container below with a centered, cone shaped hole drilled scale.) use 3.1416 for the approximate value of π. round your answer to two decimal places. determine the side length and volume of the cube. s = 12 12 ≈ 12 m v_cube = m³ determine the radius, height, and volume of the cone. r = m h = m v_cone = m³ determine the volume of the container after the cone has been removed. v_total = m³

Explanation:

Volume of the Cube

Step1: Identify side length of cube

The side length \( s \) of the cube is given as \( 12 \, m \) (from the diagram, the diameter of the cone's base is \( 6 \, m \), but the cube's side is twice that? Wait, no, the cube's side length: looking at the cone's diameter is \( 6 \, m \), but the cube's side—wait, the problem says \( s = 12 \, m \) (as per the given box). So side length \( s = 12 \, m \).

Step2: Calculate volume of cube

The formula for the volume of a cube is \( V_{\text{cube}} = s^3 \).
Substituting \( s = 12 \, m \):
\( V_{\text{cube}} = 12^3 = 12 \times 12 \times 12 = 1728 \, m^3 \).

Step1: Find radius of cone

The diameter of the cone's base is \( 6 \, m \) (from the diagram). Radius \( r = \frac{\text{diameter}}{2} = \frac{6}{2} = 3 \, m \).

Step2: Find height of cone

The height of the cone is equal to the side length of the cube (since the hole is drilled through the cube), so \( h = 12 \, m \).

Step3: Calculate volume of cone

The formula for the volume of a cone is \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h \).
Substituting \( r = 3 \, m \), \( h = 12 \, m \), and \( \pi = 3.1416 \):
\( V_{\text{cone}} = \frac{1}{3} \times 3.1416 \times 3^2 \times 12 \)
First, calculate \( 3^2 = 9 \):
\( V_{\text{cone}} = \frac{1}{3} \times 3.1416 \times 9 \times 12 \)
Simplify \( \frac{1}{3} \times 9 = 3 \):
\( V_{\text{cone}} = 3.1416 \times 3 \times 12 \)
\( 3 \times 12 = 36 \):
\( V_{\text{cone}} = 3.1416 \times 36 = 113.0976 \, m^3 \)

Step1: Subtract cone volume from cube volume

The total volume \( V_{\text{total}} = V_{\text{cube}} - V_{\text{cone}} \).
We know \( V_{\text{cube}} = 1728 \, m^3 \) and \( V_{\text{cone}} = 113.0976 \, m^3 \).
\( V_{\text{total}} = 1728 - 113.0976 = 1614.9024 \, m^3 \)
Round to two decimal places: \( 1614.90 \, m^3 \).

Answer:

\( V_{\text{cube}} = \boldsymbol{1728} \, m^3 \)

Radius, Height, and Volume of the Cone