Question
Dylan 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: 18 in, 10 in and 15 in
Dylan's aquarium
cubic inches of water can fit in the aquarium.Decompose into rectangles, then add the area of each rectangle.
Solve on paper, then enter your answer on Zearn. You can use the calculator to help you solve.
Triangular prism with side lengths labeled: 18 in, 10 in and 15 in
Dylan's aquarium
cubic inches of water can fit in the aquarium.Decompose into rectangles, then add the area of each rectangle.
Answers
GPT-4o mini
To find the volume of a triangular prism, we can follow these steps:
1. **Calculate the area of the triangular base**.
2. **Multiply the area of the base by the height of the prism**.
The triangular base has side lengths of 18 inches, 10 inches, and 15 inches. We can use Heron's formula to find the area of the triangle.
### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated as follows:
\[
s = \frac{a + b + c}{2}
\]
where \( a = 18 \) in, \( b = 10 \) in, and \( c = 15 \) in.
\[
s = \frac{18 + 10 + 15}{2} = \frac{43}{2} = 21.5 \text{ in}
\]
### Step 2: Calculate the area (A) using Heron's formula
Heron's formula states:
\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Now substituting for \( s \), \( a \), \( b \), and \( c \):
\[
A = \sqrt{21.5(21.5 - 18)(21.5 - 10)(21.5 - 15)}
\]
Calculating inside the parentheses:
\[
s - a = 21.5 - 18 = 3.5
\]
\[
s - b = 21.5 - 10 = 11.5
\]
\[
s - c = 21.5 - 15 = 6.5
\]
Now we can substitute these values back into the formula for the area:
\[
A = \sqrt{21.5 \cdot 3.5 \cdot 11.5 \cdot 6.5}
\]
### Step 3: Calculate the area
Calculating the product:
\[
21.5 \cdot 3.5 = 75.25
\]
\[
11.5 \cdot 6.5 = 74.75
\]
Now calculate the product of these two results:
\[
75.25 \cdot 74.75 = 5625.6875
\]
Taking the square root:
\[
A = \sqrt{5625.6875} \approx 75 \text{ sq in}
\]
### Step 4: Calculate the volume of the prism
Now that we have the area of the triangular base, we need to multiply it by the height of the prism. Here, we need to know the height. If the height \( h \) of the prism that represents how tall the aquarium is, let’s assume it is 15 inches (the last dimension not used).
So, the volume \( V \) is calculated as:
\[
V = A \cdot h = 75 \cdot 15 = 1125 \text{ cubic inches}
\]
Thus, the volume of water that can fit in the aquarium is:
\[
\text{1125 cubic inches}
\]
Enter this answer into Zearn:
\[
\text{1125}
\]
1. **Calculate the area of the triangular base**.
2. **Multiply the area of the base by the height of the prism**.
The triangular base has side lengths of 18 inches, 10 inches, and 15 inches. We can use Heron's formula to find the area of the triangle.
### Step 1: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated as follows:
\[
s = \frac{a + b + c}{2}
\]
where \( a = 18 \) in, \( b = 10 \) in, and \( c = 15 \) in.
\[
s = \frac{18 + 10 + 15}{2} = \frac{43}{2} = 21.5 \text{ in}
\]
### Step 2: Calculate the area (A) using Heron's formula
Heron's formula states:
\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Now substituting for \( s \), \( a \), \( b \), and \( c \):
\[
A = \sqrt{21.5(21.5 - 18)(21.5 - 10)(21.5 - 15)}
\]
Calculating inside the parentheses:
\[
s - a = 21.5 - 18 = 3.5
\]
\[
s - b = 21.5 - 10 = 11.5
\]
\[
s - c = 21.5 - 15 = 6.5
\]
Now we can substitute these values back into the formula for the area:
\[
A = \sqrt{21.5 \cdot 3.5 \cdot 11.5 \cdot 6.5}
\]
### Step 3: Calculate the area
Calculating the product:
\[
21.5 \cdot 3.5 = 75.25
\]
\[
11.5 \cdot 6.5 = 74.75
\]
Now calculate the product of these two results:
\[
75.25 \cdot 74.75 = 5625.6875
\]
Taking the square root:
\[
A = \sqrt{5625.6875} \approx 75 \text{ sq in}
\]
### Step 4: Calculate the volume of the prism
Now that we have the area of the triangular base, we need to multiply it by the height of the prism. Here, we need to know the height. If the height \( h \) of the prism that represents how tall the aquarium is, let’s assume it is 15 inches (the last dimension not used).
So, the volume \( V \) is calculated as:
\[
V = A \cdot h = 75 \cdot 15 = 1125 \text{ cubic inches}
\]
Thus, the volume of water that can fit in the aquarium is:
\[
\text{1125 cubic inches}
\]
Enter this answer into Zearn:
\[
\text{1125}
\]