Find the volume of a right triangular prism in which the height is 100 cm and which base is a right triangle with a hypotenuse 57cm in length and an angle of 43 degrees.



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

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

### Step 1: Find the area of the triangular base
The triangular base is a right triangle with:
- Hypotenuse \( c = 57 \, \text{cm} \)
- An angle \( \theta = 43^\circ \)

To find the area of the triangle, we need the lengths of the two legs (\( a \) and \( b \)). Using trigonometry, we can express the legs in terms of \( c \) and \( \theta \):

\[
a = c \cdot \cos(\theta)
\]
\[
b = c \cdot \sin(\theta)
\]

#### Calculations:
1. **Calculate \( a \)** (the side adjacent to \( \theta \)):
   \[
   a = 57 \cdot \cos(43^\circ)
   \]

2. **Calculate \( b \)** (the side opposite \( \theta \)):
   \[
   b = 57 \cdot \sin(43^\circ)
   \]

3. **Calculate the area of the triangle**:
   \[
   \text{Area of triangle} = \frac{1}{2} \cdot a \cdot b
   \]

### Step 2: Find the volume of the prism
Once we have the area of the base triangle, we multiply by the prism's height (100 cm):

\[
V = \text{Base Area} \times 100
\]

Let’s compute each part to find the answer.1. **Lengths of the legs**:
   - \( a \approx 41.69 \, \text{cm} \)
   - \( b \approx 38.87 \, \text{cm} \)

2. **Area of the triangular base**:
   \[
   \text{Area} \approx 810.27 \, \text{cm}^2
   \]

3. **Volume of the prism**:
   \[
   V \approx 81027.14 \, \text{cm}^3
   \]

Thus, the volume of the prism is approximately **81,027.14 cm³**.

Would you like further details or have any questions?

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### Related Questions:
1. How do you find the volume of a prism with a non-right triangle as a base?
2. What changes if the prism height was doubled but the base remained the same?
3. How would you calculate the volume if the angle given was different?
4. What if the hypotenuse was the unknown side—how would that change calculations?
5. How do you find the surface area of this prism?

### Tip:
When working with trigonometric functions, ensure your calculator or software is set to the correct mode (degrees or radians) to avoid errors.

Find the volume of a right triangular prism in which the height is 100 cm and which base is a right triangle with a hypotenuse 57 cm in length and an angle of 43 degrees.

Explore how to calculate the volume of a right triangular prism with a height of 100 cm, a base with a 57 cm hypotenuse, and a 43° angle. This solution involves trigonometric calculations to find the leg lengths and the area of the triangle. Suitable for high school students studying geometry and trigonometry.

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