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Question

An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side of the triangle measures 10 centimeters. The right side of the triangle also measures 10 centimeters. The perpendicular height from the top vertex of the triangle to the base measures 6 centimeters. The base of the triangle measures 8 centimeters.

What is the missing length L of the triangular prism if the surface area is 524 square centimeters?

Responses

8 cm

16 cm

10 cm

17 cm

Answer 1

To find the missing length \( L \) of the triangular prism, we first need to calculate the area of the triangular face and then use the surface area formula of the prism.

1. Calculate the area of the triangular face: The area \( A \) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Given:

   • Base \( = 8 \) cm
   • Height \( = 6 \) cm

   Substituting the values: \[ A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2 \]

2. Calculate the surface area of the prism: The surface area \( S \) of a triangular prism can be calculated using the formula: \[ S = (2 \times \text{Area of triangular face}) + \text{Perimeter of triangular face} \times L \] The perimeter \( P \) of the triangular face is the sum of all its sides: \[ P = 10 + 10 + 8 = 28 \text{ cm} \]

   Therefore, substituting the values we have: \[ S = 2 \times 24 + 28 \times L \] Simplifying this: \[ S = 48 + 28L \]

3. Set the surface area equal to 524 cm² and solve for \( L \): \[ 48 + 28L = 524 \] Subtract 48 from both sides: \[ 28L = 524 - 48 \] \[ 28L = 476 \] Divide both sides by 28: \[ L = \frac{476}{28} = 17 \]

Thus, the missing length \( L \) of the triangular prism is 17 cm.