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Volume of Composite 3D Figures Practice
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4 of 54 of 5 Items
Question
Use the image to answer the question.
An illustration shows a hexagonal prism with the label Choosy Cheesy on the top surface. The height of the hexagonal prism is 8 centimeters. The two longer parallel sides of the hexagonal prism each measure 30 centimeters in length. One end of the prism is marked with a dashed line measuring 15 centimeters across, joining the two opposite vertices of the parallel edges, forming a triangular prism with a perpendicular measure of 8 centimeters.
A sample of gourmet cheeses came in this box. What is the volume of the box?
(1 point)
Volume of Composite 3D Figures Practice
Complete this assessment to review what you've learned. It will not count toward your grade.
4 of 54 of 5 Items
Question
Use the image to answer the question.
An illustration shows a hexagonal prism with the label Choosy Cheesy on the top surface. The height of the hexagonal prism is 8 centimeters. The two longer parallel sides of the hexagonal prism each measure 30 centimeters in length. One end of the prism is marked with a dashed line measuring 15 centimeters across, joining the two opposite vertices of the parallel edges, forming a triangular prism with a perpendicular measure of 8 centimeters.
A sample of gourmet cheeses came in this box. What is the volume of the box?
(1 point)
Answers
Answered by
GPT-4o mini
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:**
1. The formula for the volume of a hexagonal prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
2. 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:
\[
\text{Area} = \frac{\sqrt{3}}{4} a^2
\]
3. 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:
\[
a = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ cm}
\]
4. The area of the hexagon, considering there are 6 triangles, is:
\[
\text{Base Area} = 6 \times \frac{\sqrt{3}}{4} a^2 = 6 \times \frac{\sqrt{3}}{4} (5\sqrt{3})^2
\]
\[
= 6 \times \frac{\sqrt{3}}{4} (75) = 6 \times \frac{75\sqrt{3}}{4} = \frac{450\sqrt{3}}{4} = 112.5\sqrt{3}
\]
5. 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:**
1. The volume of a triangular prism is calculated using:
\[
V = \text{Base Area} \times \text{Height}
\]
2. 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):
\[
\text{Base Area} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 15 \times 8 = 60 \text{ cm}^2
\]
3. 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.
**Hexagonal Prism:**
1. The formula for the volume of a hexagonal prism is:
\[
V = \text{Base Area} \times \text{Height}
\]
2. 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:
\[
\text{Area} = \frac{\sqrt{3}}{4} a^2
\]
3. 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:
\[
a = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ cm}
\]
4. The area of the hexagon, considering there are 6 triangles, is:
\[
\text{Base Area} = 6 \times \frac{\sqrt{3}}{4} a^2 = 6 \times \frac{\sqrt{3}}{4} (5\sqrt{3})^2
\]
\[
= 6 \times \frac{\sqrt{3}}{4} (75) = 6 \times \frac{75\sqrt{3}}{4} = \frac{450\sqrt{3}}{4} = 112.5\sqrt{3}
\]
5. 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:**
1. The volume of a triangular prism is calculated using:
\[
V = \text{Base Area} \times \text{Height}
\]
2. 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):
\[
\text{Base Area} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 15 \times 8 = 60 \text{ cm}^2
\]
3. 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.