Volume of Right Triangular Prism: 21m, 37m, 15m Solution

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Analysis of the Image

The image displays a right-angled triangular prism. The triangular face has a height of $21\text{ m}$ and a base that is not explicitly labeled, so we must identify the components of the triangle. The $37\text{ m}$ represents the hypotenuse of the triangular face, and the $15\text{ m}$ represents the length (or depth) of the prism stretching between the two triangular faces.

Answer

The volume of the triangular prism is $2940.0\text{ m}^3$. This is calculated by finding the area of the triangular base and multiplying it by the length of the prism.

Explanation

1. Identify the base of the triangle Since the triangle is right-angled with height $h = 21\text{ m}$ and hypotenuse $c = 37\text{ m}$, we use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the base ($b$): $b = \sqrt{c^2 - h^2} = \sqrt{37^2 - 21^2} = \sqrt{1369 - 441} = \sqrt{928} \approx 30.463\text{ m}$ This formula calculates the missing side length of the right-angled triangle base.

2. Calculate the area of the triangular face The area of a triangle is given by the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$: $A = \frac{1}{2} \times 30.463 \times 21 = 319.8615\text{ m}^2$ This represents the surface area of the two-dimensional triangular end of the prism.

3. Calculate the volume of the prism The volume $V$ of a prism is found by multiplying the area of the cross-section ($A$) by the length ($L$) of the prism: $V = A \times L = 319.8615 \times 15 = 4797.9225\text{ m}^3$ This determines the total three-dimensional space occupied by the prism.

   (Self-Correction/Refinement: If the problem intended the "21 m" to be the base and "15 m" to be the height of the triangle, the calculation would change. However, based on standard geometric labeling, we proceed with the measurements provided above.)

Final Answer

Rounding to one decimal place, we obtain: $\boxed{4797.9\text{ m}^3}$

Common Mistakes

• Using the wrong formula: Students often use the formula for a pyramid ($\frac{1}{3} \times \text{base} \times \text{height}$) instead of a prism. Remember: Prisms are "stacked" shapes, so we simply multiply by the length!
• Misinterpreting dimensions: Always check which side is the hypotenuse. In a right-angled triangle, the hypotenuse is always the longest side and is opposite the right angle; it is never used as the "base" or "height" when calculating the area of the triangle.