Question
Corey built an aquarium. How many cubic inches of water can fit in the aquarium?
Solve on paper, then enter your answer on Zearn. You can use the calculator to help you solve.
Triangular prism with side lengths labeled: 15 in, 12 in and 10 in.
Corey's aquarium
cubic inches of water can fit in the aquarium.
Solve on paper, then enter your answer on Zearn. You can use the calculator to help you solve.
Triangular prism with side lengths labeled: 15 in, 12 in and 10 in.
Corey's aquarium
cubic inches of water can fit in the aquarium.
Answers
GPT-4o mini
To find the volume of a triangular prism, we can use the formula:
\[
\text{Volume} = \text{Area of the base} \times \text{Height}
\]
1. **Find the area of the triangular base.** For a triangle, we can use Heron's formula, which requires the semi-perimeter and the lengths of the sides.
The sides are:
- \(a = 15\) in
- \(b = 12\) in
- \(c = 10\) in
First, calculate the semi-perimeter \(s\):
\[
s = \frac{a + b + c}{2} = \frac{15 + 12 + 10}{2} = \frac{37}{2} = 18.5 \text{ in}
\]
2. **Now apply Heron’s formula:**
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
Calculating each term:
- \(s - a = 18.5 - 15 = 3.5\)
- \(s - b = 18.5 - 12 = 6.5\)
- \(s - c = 18.5 - 10 = 8.5\)
Now substituting into Heron's formula:
\[
\text{Area} = \sqrt{18.5 \cdot 3.5 \cdot 6.5 \cdot 8.5}
\]
Calculating the products:
\[
18.5 \cdot 3.5 = 64.75
\]
\[
6.5 \cdot 8.5 = 55.25
\]
Now multiply those results:
\[
64.75 \cdot 55.25 \approx 3587.1875
\]
Taking the square root:
\[
\text{Area} \approx \sqrt{3587.1875} \approx 59.9 \text{ square inches}
\]
3. **Now, multiply by the height (which is the length of the prism).** We need to know the height of the prism. If not specified, assume the height is given as an additional dimension or a context clue in the problem.
Let's assume Corey’s aquarium height (length of the prism) is \(h\) (not provided, so I will use \(h\) in my final formula):
4. **Finally, the volume of the aquarium (triangular prism) will be:**
\[
\text{Volume} = \text{Area} \times h = 59.9 \times h \text{ cubic inches}
\]
If \(h\) is known, substitute it in to get the volume; otherwise, express in terms of \(h\).
Make sure to enter the final volume calculation based on the height of the aquarium. If you find the height to be, for example, 10 inches, you would compute \(59.9 \times 10 = 599\) cubic inches.
Please provide the prism's height for a specific answer, or you can complete this calculation based on the height provided in your assignment.
\[
\text{Volume} = \text{Area of the base} \times \text{Height}
\]
1. **Find the area of the triangular base.** For a triangle, we can use Heron's formula, which requires the semi-perimeter and the lengths of the sides.
The sides are:
- \(a = 15\) in
- \(b = 12\) in
- \(c = 10\) in
First, calculate the semi-perimeter \(s\):
\[
s = \frac{a + b + c}{2} = \frac{15 + 12 + 10}{2} = \frac{37}{2} = 18.5 \text{ in}
\]
2. **Now apply Heron’s formula:**
\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
Calculating each term:
- \(s - a = 18.5 - 15 = 3.5\)
- \(s - b = 18.5 - 12 = 6.5\)
- \(s - c = 18.5 - 10 = 8.5\)
Now substituting into Heron's formula:
\[
\text{Area} = \sqrt{18.5 \cdot 3.5 \cdot 6.5 \cdot 8.5}
\]
Calculating the products:
\[
18.5 \cdot 3.5 = 64.75
\]
\[
6.5 \cdot 8.5 = 55.25
\]
Now multiply those results:
\[
64.75 \cdot 55.25 \approx 3587.1875
\]
Taking the square root:
\[
\text{Area} \approx \sqrt{3587.1875} \approx 59.9 \text{ square inches}
\]
3. **Now, multiply by the height (which is the length of the prism).** We need to know the height of the prism. If not specified, assume the height is given as an additional dimension or a context clue in the problem.
Let's assume Corey’s aquarium height (length of the prism) is \(h\) (not provided, so I will use \(h\) in my final formula):
4. **Finally, the volume of the aquarium (triangular prism) will be:**
\[
\text{Volume} = \text{Area} \times h = 59.9 \times h \text{ cubic inches}
\]
If \(h\) is known, substitute it in to get the volume; otherwise, express in terms of \(h\).
Make sure to enter the final volume calculation based on the height of the aquarium. If you find the height to be, for example, 10 inches, you would compute \(59.9 \times 10 = 599\) cubic inches.
Please provide the prism's height for a specific answer, or you can complete this calculation based on the height provided in your assignment.