Question
Use betta 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?
cm3
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?
cm3
Answers
GPT-4o mini
To find the volume of the hexagonal prism labeled "Choosy Cheesy", we first need to calculate the area of the hexagonal base and then multiply it by the height of the prism.
**Step 1: Calculate the area of the hexagonal base.**
The hexagonal prism has two longer parallel sides that measure 30 centimeters each. The dashed line measuring 15 centimeters connects two opposite vertices of the hexagon. The dashed line refers to the distance across the hexagon — this is known as the width of the hexagon along one of its diagonals.
A regular hexagon can be divided into 6 equilateral triangles. Knowing the distance across opposite vertices (the diameter of the circumscribed circle) can help us understand more about the hexagon's properties.
1. **Find the radius (R) of the circumscribed circle:**
The diameter across opposite vertices (d) is given as 15 cm. Thus, the radius \( R \) of the circle is:
\[
R = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ cm}
\]
2. **Find the side length (s) of the hexagon:**
The radius of the circle to which the hexagon is inscribed also represents the distance from the center of the hexagon to each vertex (this distance is equal to the side length in a regular hexagon). Therefore, the side length \( s \) of the hexagon is also 7.5 cm.
3. **Calculate the area (A) of the hexagonal base:**
The area \( A \) of a regular hexagon can be calculated using the formula:
\[
A = \frac{3\sqrt{3}}{2}s^2
\]
Substituting \( s = 7.5 \):
\[
A = \frac{3\sqrt{3}}{2} \times (7.5)^2
\]
\[
= \frac{3\sqrt{3}}{2} \times 56.25
\]
\[
= \frac{3\sqrt{3} \times 56.25}{2} \approx \frac{3 \times 1.732 \times 56.25}{2} \approx \frac{291.285}{2} \approx 145.6425 \text{ cm}^2
\]
**Step 2: Calculate the volume of the prism (V).**
The volume \( V \) of the prism is calculated as:
\[
V = A \times h
\]
Where \( h \) is the height of the prism (8 cm):
\[
V = 145.6425 \times 8
\]
\[
= 1165.14 \, \text{cm}^3
\]
Thus, the volume of the box is approximately **1165.14 cm³**.
**Step 1: Calculate the area of the hexagonal base.**
The hexagonal prism has two longer parallel sides that measure 30 centimeters each. The dashed line measuring 15 centimeters connects two opposite vertices of the hexagon. The dashed line refers to the distance across the hexagon — this is known as the width of the hexagon along one of its diagonals.
A regular hexagon can be divided into 6 equilateral triangles. Knowing the distance across opposite vertices (the diameter of the circumscribed circle) can help us understand more about the hexagon's properties.
1. **Find the radius (R) of the circumscribed circle:**
The diameter across opposite vertices (d) is given as 15 cm. Thus, the radius \( R \) of the circle is:
\[
R = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ cm}
\]
2. **Find the side length (s) of the hexagon:**
The radius of the circle to which the hexagon is inscribed also represents the distance from the center of the hexagon to each vertex (this distance is equal to the side length in a regular hexagon). Therefore, the side length \( s \) of the hexagon is also 7.5 cm.
3. **Calculate the area (A) of the hexagonal base:**
The area \( A \) of a regular hexagon can be calculated using the formula:
\[
A = \frac{3\sqrt{3}}{2}s^2
\]
Substituting \( s = 7.5 \):
\[
A = \frac{3\sqrt{3}}{2} \times (7.5)^2
\]
\[
= \frac{3\sqrt{3}}{2} \times 56.25
\]
\[
= \frac{3\sqrt{3} \times 56.25}{2} \approx \frac{3 \times 1.732 \times 56.25}{2} \approx \frac{291.285}{2} \approx 145.6425 \text{ cm}^2
\]
**Step 2: Calculate the volume of the prism (V).**
The volume \( V \) of the prism is calculated as:
\[
V = A \times h
\]
Where \( h \) is the height of the prism (8 cm):
\[
V = 145.6425 \times 8
\]
\[
= 1165.14 \, \text{cm}^3
\]
Thus, the volume of the box is approximately **1165.14 cm³**.