Composite Solid Volume Calculation

The problem asks for the total volume of a composite solid formed by placing a pyramid on top of a cube.

The total volume is the sum of the volume of the cube and the volume of the pyramid.

V_total = V_cube + V_pyramid

Volume of the Cube

The cube has a side length s = 10 cm.

The formula for the volume of a cube is:

V_cube = s³

Substituting the value:

V_cube = (10 \text{ cm})^3 = 1000 \text{ cm}^3

Volume of the Pyramid

The pyramid has an equilateral triangular base with side length a = 10 cm and a height h_pyramid = 14 cm.

First, calculate the area of the equilateral triangular base (Base Area):

\text{Base Area} = \frac{\sqrt{3}}{4} a^2

\text{Base Area} = \frac{\sqrt{3}}{4} (10 \text{ cm})^2 = \frac{\sqrt{3}}{4} \times 100 \text{ cm}^2 = 25\sqrt{3} \text{ cm}^2

Using the approximation $\sqrt{3} \approx 1.73205$:

\text{Base Area} \approx 25 \times 1.73205 \approx 43.301 \text{ cm}^2

The formula for the volume of a pyramid is:

V_pyramid = \frac{1}{3} \times \text{Base Area} \times h_{pyramid}

Substituting the values:

V_{pyramid} = \frac{1}{3} \times (25\sqrt{3} \text{ cm}^2) \times (14 \text{ cm})}

V_{pyramid} = \frac{350\sqrt{3}}{3} \text{ cm}^3}

Calculating the approximate value:

V_{pyramid} \approx \frac{350 \times 1.73205}{3} \approx \frac{606.2175}{3} \approx 202.07 \text{ cm}^3}

Total Volume Calculation

Add the volume of the cube and the volume of the pyramid:

V_total = V_cube + V_pyramid

V_{total} \approx 1000 \text{ cm}^3 + 202.07 \text{ cm}^3}

V_{total} \approx 1202.07 \text{ cm}^3}