Question

A solid composite consisting of a cube edge 28cm and a square based pyramid of height 16cm calculate the volume of the solid

Answer 1

To calculate the volume of the composite solid consisting of a cube and a square-based pyramid, we'll compute the volume of each component separately and then add them together.

### Step 1: Volume of the Cube
The formula for the volume of a cube is given by:

\[
V_{\text{cube}} = a^3
\]

where \(a\) is the length of an edge of the cube.

Given that the edge of the cube is \(28\) cm:

\[
V_{\text{cube}} = 28^3 = 28 \times 28 \times 28
\]
Calculating this:

\[
28 \times 28 = 784
\]
\[
784 \times 28 = 21952 \text{ cm}^3
\]

So, the volume of the cube is \(21952 \text{ cm}^3\).

### Step 2: Volume of the Square-Based Pyramid
The formula for the volume of a square-based pyramid is given by:

\[
V_{\text{pyramid}} = \frac{1}{3} \times B \times h
\]

where \(B\) is the area of the base, and \(h\) is the height of the pyramid.

For a pyramid with a square base, the area \(B\) of the base can be calculated as:

\[
B = a^2
\]

Given that the base of the pyramid also has an edge length equal to the edge of the cube, which is \(28\) cm:

\[
B = 28^2 = 784 \text{ cm}^2
\]

Now, the height \(h\) of the pyramid is given as \(16\) cm:

\[
V_{\text{pyramid}} = \frac{1}{3} \times 784 \times 16
\]

Calculating this:

First, calculate \(784 \times 16\):

\[
784 \times 16 = 12544
\]

Now, divide by 3 to find the volume of the pyramid:

\[
V_{\text{pyramid}} = \frac{12544}{3} \approx 4181.33 \text{ cm}^3
\]

### Step 3: Total Volume of the Composite Solid
Now, add the volumes of the cube and the pyramid:

\[
V_{\text{total}} = V_{\text{cube}} + V_{\text{pyramid}} = 21952 + 4181.33
\]

Calculating this:

\[
V_{\text{total}} \approx 26133.33 \text{ cm}^3
\]

### Final Answer
The total volume of the composite solid is approximately \(26133.33 \text{ cm}^3\).