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?
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 can use the formula for the volume of a prism:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
### Step 1: Determine the area of the hexagonal base
The hexagonal prism's base is a hexagon. A regular hexagon can be divided into 6 equilateral triangles. The area \(A\) of a regular hexagon can also be calculated using the following formula:
\[
A = \frac{3\sqrt{3}}{2} s^2
\]
where \(s\) is the length of a side of the hexagon.
### Step 2: Find the side length of the hexagon
From the description, we see that the hexagon has longer parallel sides measuring 30 cm and the dashed line measuring 15 cm across. Since the dashed line connects opposite vertices, it represents the width between two parallel vertices.
For a regular hexagon, if the distance between two opposite sides (the distance across the hexagon) is \(d\), the relationship is given by:
\[
d = s \sqrt{3}
\]
Since the two longer sides are each 30 cm, we can assume the side length \(s\) relates to this. The total distance between two parallel sides (30 cm) would reflect the diameter of the inscribed circle of the hexagon given the length of its side.
However, based on the description (15 cm joining opposite vertices), let's find \(s\) from the width of the regular hexagon. The distance across the hexagon at 15 cm matches the arrangement, so we can align that correctly into the formula we derived:
Given that \(d\) is 15 cm, we utilize:
\[
s \sqrt{3} = 15 \implies s = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ cm}
\]
### Step 3: Calculate the area of the hexagon \(A\)
Substituting \(s\) back into the area formula:
\[
A = \frac{3\sqrt{3}}{2} (5\sqrt{3})^2 = \frac{3\sqrt{3}}{2} \cdot 75 = \frac{225\sqrt{3}}{2} \text{ cm}^2
\]
### Step 4: Calculate the volume
Now, substituting \(A\) into the volume formula and considering the height \(h = 8\) cm:
\[
\text{Volume} = A \times h = \frac{225\sqrt{3}}{2} \times 8
\]
This calculation gives:
\[
= \frac{225 \times 8 \sqrt{3}}{2} = 900\sqrt{3} \text{ cm}^3
\]
### Final Volume
Thus, the volume of the box is:
\[
\text{Volume} \approx 900 \times 1.732 \approx 1550.45 \text{ cm}^3
\]
Therefore, the volume of the box is approximately:
\[
\boxed{900\sqrt{3} \text{ or } 1550.45 \text{ cm}^3}
\]
Please feel free to ask if you have further questions or clarifications!
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
### Step 1: Determine the area of the hexagonal base
The hexagonal prism's base is a hexagon. A regular hexagon can be divided into 6 equilateral triangles. The area \(A\) of a regular hexagon can also be calculated using the following formula:
\[
A = \frac{3\sqrt{3}}{2} s^2
\]
where \(s\) is the length of a side of the hexagon.
### Step 2: Find the side length of the hexagon
From the description, we see that the hexagon has longer parallel sides measuring 30 cm and the dashed line measuring 15 cm across. Since the dashed line connects opposite vertices, it represents the width between two parallel vertices.
For a regular hexagon, if the distance between two opposite sides (the distance across the hexagon) is \(d\), the relationship is given by:
\[
d = s \sqrt{3}
\]
Since the two longer sides are each 30 cm, we can assume the side length \(s\) relates to this. The total distance between two parallel sides (30 cm) would reflect the diameter of the inscribed circle of the hexagon given the length of its side.
However, based on the description (15 cm joining opposite vertices), let's find \(s\) from the width of the regular hexagon. The distance across the hexagon at 15 cm matches the arrangement, so we can align that correctly into the formula we derived:
Given that \(d\) is 15 cm, we utilize:
\[
s \sqrt{3} = 15 \implies s = \frac{15}{\sqrt{3}} = 5\sqrt{3} \text{ cm}
\]
### Step 3: Calculate the area of the hexagon \(A\)
Substituting \(s\) back into the area formula:
\[
A = \frac{3\sqrt{3}}{2} (5\sqrt{3})^2 = \frac{3\sqrt{3}}{2} \cdot 75 = \frac{225\sqrt{3}}{2} \text{ cm}^2
\]
### Step 4: Calculate the volume
Now, substituting \(A\) into the volume formula and considering the height \(h = 8\) cm:
\[
\text{Volume} = A \times h = \frac{225\sqrt{3}}{2} \times 8
\]
This calculation gives:
\[
= \frac{225 \times 8 \sqrt{3}}{2} = 900\sqrt{3} \text{ cm}^3
\]
### Final Volume
Thus, the volume of the box is:
\[
\text{Volume} \approx 900 \times 1.732 \approx 1550.45 \text{ cm}^3
\]
Therefore, the volume of the box is approximately:
\[
\boxed{900\sqrt{3} \text{ or } 1550.45 \text{ cm}^3}
\]
Please feel free to ask if you have further questions or clarifications!