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 \(26133.33 \text{ cm}^3\).
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