https://file.aiquickdraw.com/mathbot/file/17292843597834o77hk62.jpg

To calculate the volume of a triangular prism, we use the formula:

\[
V = \text{Base Area} \times \text{Height of the prism}
\]

In this problem, we need to find the area of the triangular base and then multiply it by the height of the prism (which extends between the parallel triangular faces).

### Step 1: Calculate the Area of the Triangular Base
The triangular base is defined by sides \( p \), \( q \), and the angle \( H \) between them. The formula for the area of a triangle when two sides and the included angle are given is:

\[
\text{Area} = \frac{1}{2} \times p \times q \times \sin(H)
\]

From the image:
- \( p = 7 \, \text{cm} \)
- \( q = \sqrt{17} \, \text{cm} \)
- \( H = 51.34^\circ \)

### Step 2: Convert Angle to Radians
Since trigonometric functions often require radians in calculators:

\[
H = 51.34^\circ = 51.34 \times \frac{\pi}{180} \, \text{radians}
\]

### Step 3: Calculate the Area of the Triangular Base
\[
\text{Area} = \frac{1}{2} \times 7 \times \sqrt{17} \times \sin(51.34^\circ)
\]

### Step 4: Volume of the Triangular Prism
Multiply the base area by the height of the prism, which is given as 7 cm:

\[
V = \text{Base Area} \times 7
\]

Let's calculate this step-by-step.The volume of the triangular prism is approximately \( 79 \, \text{cm}^3 \) when rounded to the nearest whole number.

Would you like more details or have any questions?

Here are some related questions for further understanding:
1. How do you find the area of a triangle with known side lengths?
2. What is the importance of converting angles to radians in trigonometric calculations?
3. How does changing the height of the prism affect its volume?
4. What would happen to the volume if the angle \( H \) changed to a different value?
5. How can you verify the calculations using a different method?

**Tip:** When working with angles in trigonometry, make sure your calculator is set to the correct mode (degrees or radians) to avoid mistakes.

Calculate the volume of the triangular prism. Give your answer to the nearest whole number. Given information: H = 51.34°.

Learn how to calculate the volume of a triangular prism using trigonometry. In this problem, the base area is calculated using two sides and the included angle, followed by multiplying the base area by the prism's height. Step-by-step solution included.

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