Question
A slice of pizza is held in a container shaped like a triangular prism. The dimensions of the container are shown in the diagram.
12 in.
7 in.
3 in.
9 in.
What is the volume of the container in cubic
12 in.
7 in.
3 in.
9 in.
What is the volume of the container in cubic
Answers
GPT-4o mini
To calculate the volume of the triangular prism that holds the pizza slice, we need to apply the formula for the volume of a prism:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
**Step 1: Find the area of the triangular base.**
From the dimensions provided, let’s identify the base of the triangle. We have three sides of the triangle with lengths of 12 inches, 7 inches, and 9 inches.
We can use Heron's formula for the area of a triangle, which is given by:
\[
s = \frac{a + b + c}{2}
\]
where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
Calculating the semi-perimeter \(s\):
\[
s = \frac{12 + 7 + 9}{2} = \frac{28}{2} = 14
\]
Now, we apply Heron's formula for the area \(A\):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Substituting the values:
\[
A = \sqrt{14(14-12)(14-7)(14-9)}
\]
Calculating each term:
\[
s - a = 14 - 12 = 2
\]
\[
s - b = 14 - 7 = 7
\]
\[
s - c = 14 - 9 = 5
\]
Now substitute these back into the area formula:
\[
A = \sqrt{14 \times 2 \times 7 \times 5}
\]
Calculating inside the square root:
\[
A = \sqrt{14 \times 2 = 28}
\]
\[
28 \times 7 = 196
\]
\[
196 \times 5 = 980
\]
Thus,
\[
A = \sqrt{980}
\]
To simplify \(\sqrt{980}\), we find the prime factorization of 980:
\[
980 = 4 \times 245 = 4 \times 5 \times 49 = 4 \times 5 \times 7^2
\]
Thus:
\[
\sqrt{980} = \sqrt{4 \times 5 \times 49} = \sqrt{4} \times \sqrt{5} \times \sqrt{49} = 2 \times 7 \times \sqrt{5} = 14\sqrt{5} \text{ in}^2
\]
**Step 2: Find the volume of the prism.**
Next, we need the height of the prism, which is given as 3 inches.
Finally, we calculate the volume:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 14\sqrt{5} \times 3 = 42\sqrt{5} \text{ in}^3
\]
Therefore, the volume of the container is:
\[
\boxed{42\sqrt{5}} \text{ cubic inches.}
\]
If a numerical approximation is needed, the approximate value can be calculated using \(\sqrt{5} \approx 2.236\):
\[
42 \times 2.236 \approx 93.192 \text{ cubic inches.}
\]
So you can also denote the volume as approximately \(93.19 \text{ cubic inches.}\)
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
**Step 1: Find the area of the triangular base.**
From the dimensions provided, let’s identify the base of the triangle. We have three sides of the triangle with lengths of 12 inches, 7 inches, and 9 inches.
We can use Heron's formula for the area of a triangle, which is given by:
\[
s = \frac{a + b + c}{2}
\]
where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
Calculating the semi-perimeter \(s\):
\[
s = \frac{12 + 7 + 9}{2} = \frac{28}{2} = 14
\]
Now, we apply Heron's formula for the area \(A\):
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
Substituting the values:
\[
A = \sqrt{14(14-12)(14-7)(14-9)}
\]
Calculating each term:
\[
s - a = 14 - 12 = 2
\]
\[
s - b = 14 - 7 = 7
\]
\[
s - c = 14 - 9 = 5
\]
Now substitute these back into the area formula:
\[
A = \sqrt{14 \times 2 \times 7 \times 5}
\]
Calculating inside the square root:
\[
A = \sqrt{14 \times 2 = 28}
\]
\[
28 \times 7 = 196
\]
\[
196 \times 5 = 980
\]
Thus,
\[
A = \sqrt{980}
\]
To simplify \(\sqrt{980}\), we find the prime factorization of 980:
\[
980 = 4 \times 245 = 4 \times 5 \times 49 = 4 \times 5 \times 7^2
\]
Thus:
\[
\sqrt{980} = \sqrt{4 \times 5 \times 49} = \sqrt{4} \times \sqrt{5} \times \sqrt{49} = 2 \times 7 \times \sqrt{5} = 14\sqrt{5} \text{ in}^2
\]
**Step 2: Find the volume of the prism.**
Next, we need the height of the prism, which is given as 3 inches.
Finally, we calculate the volume:
\[
\text{Volume} = \text{Base Area} \times \text{Height} = 14\sqrt{5} \times 3 = 42\sqrt{5} \text{ in}^3
\]
Therefore, the volume of the container is:
\[
\boxed{42\sqrt{5}} \text{ cubic inches.}
\]
If a numerical approximation is needed, the approximate value can be calculated using \(\sqrt{5} \approx 2.236\):
\[
42 \times 2.236 \approx 93.192 \text{ cubic inches.}
\]
So you can also denote the volume as approximately \(93.19 \text{ cubic inches.}\)