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.

Question

What is the missing length L of the triangular prism if the surface area is 524 square centimeters?
(1 point)
Responses

10 cm

8 cm

17 cm
  
    16 cm

GPT-4o mini · Answer

To find the missing length $ L $ of the triangular prism, we'll first need to calculate the surface area of the prism provided that the surface area is 524 square centimeters. 1. **Calculate the area of the triangular base:** The formula for the area $ A $ of a triangle is: $$ A = \frac{1}{2} \times \text{base} \times \text{height} $$ Given that the base of the triangle is 8 cm and the height is 6 cm: $$ A = \frac{1}{2} \times 8 \times 6 = 24 \text{ cm}^2 $$ 2. **Calculate the lateral surface area:** A triangular prism has three rectangular lateral faces. The rectangles' dimensions are as follows: - The height of the triangle corresponds to the two rectangular faces with dimensions (10 cm, L) (for each side of the triangle). - The base of the triangle corresponds to the rectangular face with dimensions (8 cm, L). So, the areas of the rectangular faces are: - Two identical rectangles with dimensions $ 10 \text{ cm} \times L $: $$ 2 \times (10 \times L) = 20L \text{ cm}^2 $$ - One rectangle with dimensions $ 8 \text{ cm} \times L $: $$ 8L \text{ cm}^2 $$ Thus, the total lateral surface area is: $$ 20L + 8L = 28L \text{ cm}^2 $$ 3. **Calculate the total surface area of the prism:** The total surface area $ S $ of the prism is the sum of the lateral surface area and the area of the two triangular bases (twice the area of one triangle): $$ S = 28L + 2 \times \text{area of triangle} $$ Since we calculated the area of one triangle to be 24 cm²: $$ S = 28L + 2 \times 24 $$ $$ S = 28L + 48 $$ 4. **Set up the equation for the total surface area:** Given that the surface area is 524 cm²: $$ 28L + 48 = 524 $$ 5. **Solve for $ L $:** First, subtract 48 from both sides: $$ 28L = 524 - 48 $$ $$ 28L = 476 $$ Now, divide by 28: $$ L = \frac{476}{28} \approx 17 $$ Thus, the missing length $ L $ of the triangular prism is $ \boxed{17 \text{ cm}} $.