To find the missing length \( l \) of the triangular prism given that the surface area is 524 square centimeters, we will first calculate the surface area of the triangular prism using the dimensions provided.
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Calculate the area of the triangular base: The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] For this triangle:
- Base \( b = 8 , \text{cm} \)
- Height \( h = 6 , \text{cm} \)
Thus, \[ \text{Area} = \frac{1}{2} \times 8 \times 6 = 24 , \text{cm}^2 \]
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Calculate the lateral surface area: The lateral surface area of the triangular prism can be found by calculating the area of the three rectangular faces. The dimensions of these rectangles are based on the edges of the triangle and the length \( l \):
- The left rectangle (with side 10 cm) contributes: \[ \text{Area} = 10 \times l \]
- The right rectangle (also 10 cm) contributes: \[ \text{Area} = 10 \times l \]
- The base rectangle (with base 8 cm) contributes: \[ \text{Area} = 8 \times l \]
Therefore, the total lateral surface area is: \[ \text{Lateral Area} = 10l + 10l + 8l = 28l \]
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Combine the areas to get the total surface area: The total surface area \( A \) of the prism is the sum of the area of the two triangular bases and the lateral surface area: \[ \text{Total Surface Area} = 2 \times \text{Area of triangle} + \text{Lateral Area} \] Substituting the values we computed: \[ \text{Total Surface Area} = 2 \times 24 + 28l = 48 + 28l \]
We set this equal to the given surface area of 524 cm²: \[ 48 + 28l = 524 \]
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Solve for \( l \): Subtract 48 from both sides: \[ 28l = 524 - 48 \] \[ 28l = 476 \] Now, divide both sides by 28: \[ l = \frac{476}{28} = 17 \]
Thus, the missing length \( l \) of the triangular prism is 17 cm.