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Long Description (if needed) The net appears as three horizontal rectangles joined one on top of another. The sides of the middle rectangle are drawn with dashed lines. The width of the rectangles is 100 units. The height of the middle rectangle is 24 units. The middle rectangle has a right triangle adjoining its left side. The length of the hypotenuse of the triangle is 25 units and the length of its horizontal side at the top of the triangle is 7 units. An unmarked triangle the same size and shape as the other triangle adjoins the right side of the middle rectangle. The height of the top triangle is 7 units.

The net of a triangular prism is shown. What is the surface area of the triangular prism?

(1 point)
Responses

5,668 square units
5,668 square units

5,684 square units
5,684 square units

5,768 square units
5,768 square units

5,936 square units
5,936 square units
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1 answer

To determine the surface area of the triangular prism given in the description, we need to first break down the components of the prism based on the provided net.

  1. Rectangles and Triangles in the Net:

    • There are three horizontal rectangles: the bottom rectangle (base), the middle rectangle, and the top rectangle.
    • The middle rectangle (which may be the lateral face) has dimensions of width = 100 units and height = 24 units.
    • Each triangle, as described, has a right triangle joined to either side of the middle rectangle with:
      • Hypotenuse = 25 units
      • Horizontal side (top side of triangle) = 7 units

  2. Finding the Height of the Triangle:

    • We can use the Pythagorean theorem to find the height (h) of the triangle. The sides of the triangle can be calculated as: \[ h = \sqrt{(hypotenuse)^2 - (horizontal side)^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \text{ units} \]
    • Thus, each triangle has a height of 24 units.

  3. Calculating the Surface Area:

    • Area of the Triangles: Each triangle has an area given by: \[ A = \frac{1}{2} \times base \times height = \frac{1}{2} \times 7 \times 24 = \frac{1}{2} \times 168 = 84 \text{ square units} \]

    • Since there are two such triangles (one on each side of the middle rectangle), the total area of the triangles is: \[ 2 \times 84 = 168 \text{ square units} \]

    • Area of the Rectangles:

      • The bottom rectangle (the base) has an area of: \[ A = width \times height = 100 \times 24 = 2400 \text{ square units} \]
      • The middle rectangle's area is already given as part of the surface area calculation to be: \[ 100 \times 24 = 2400 \text{ square units} \]
      • The top rectangle (which is the same as the bottom rectangle) also has an area of: \[ 100 \times 7 = 700 \text{ square units} \]

    • Adding these areas together yields:

      • Surface Area = Area of Triangular Faces + Area of Bottom + Area of Sides + Area of Top \[ \text{Total Surface Area} = 2400 + 2400 + 700 + 168 = 5,668 \text{ square units} \]

Thus, the answer for the surface area of the triangular prism is:

5,668 square units