To find the volume of the composite 3D figure which consists of a hexagonal prism and a triangular prism, we will calculate the volume of each component separately and then sum them up.
Hexagonal Prism:
- The formula for the volume of a hexagonal prism is: \[ V = \text{Base Area} \times \text{Height} \]
- The base of a hexagonal prism can be divided into six equilateral triangles. The formula for the area of one equilateral triangle given the length of a side \(a\) is: \[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
- For a hexagon, where the length of each side \(a\) is half of the width across opposite vertices (diameter), which is stated to be 15 cm across, each side length will be: \[ a = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ cm} \]
- The area of the hexagon, considering there are 6 triangles, is: \[ \text{Base Area} = 6 \times \frac{\sqrt{3}}{4} a^2 = 6 \times \frac{\sqrt{3}}{4} (5\sqrt{3})^2 \] \[ = 6 \times \frac{\sqrt{3}}{4} (75) = 6 \times \frac{75\sqrt{3}}{4} = \frac{450\sqrt{3}}{4} = 112.5\sqrt{3} \]
- Now, substituting this value into the volume calculation:
- Height of the hexagonal prism = 8 cm \[ V_{hexagonal} = 112.5\sqrt{3} \times 8 = 900\sqrt{3} \text{ cm}^3 \]
Triangular Prism:
- The volume of a triangular prism is calculated using: \[ V = \text{Base Area} \times \text{Height} \]
- The base area of the triangular prism can be calculated from the triangular area:
- The base of the triangular prism is the height which is given as 8 cm, and the length is the same as the length of the two longer sides of the hexagon which is 30 cm.
- Area of the triangular base (using the height of the triangle): \[ \text{Base Area} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 15 \times 8 = 60 \text{ cm}^2 \]
- The height of the triangular prism is also 30 cm. \[ V_{triangular} = 60 \times 30 = 1800 \text{ cm}^3 \]
Total Volume of Composite Figure: \[ V_{total} = V_{hexagonal} + V_{triangular} = 900\sqrt{3} + 1800 \]
Assuming \(\sqrt{3} \approx 1.732\), the final calculation becomes:
- For the hexagonal part: \(900 \times 1.732 \approx 1558.8\)
Thus: \[ V_{total} \approx 1558.8 + 1800 \approx 3358.8 \text{ cm}^3 \]
So, the approximate volume of the box is 3358.8 cm³, but if you need to keep it in terms of exact measurements, it would be \(900\sqrt{3} + 1800\) cubic centimeters.