The two-dimensional net shows three rectangles one above the other. The length of the rectangles is 8 inches. The bottom rectangle has one right triangle on its right and one right... The two-dimensional net shows three rectangles one above the other. The length of the rectangles is 8 inches. The bottom rectangle has one right triangle on its right and one right triangle on its left. The width of the bottom rectangle and the height of the rectangle is 4 inches. The base of the right triangle is 3 inches and the slant height is 5 inches.
Understand the Problem
The question describes a two-dimensional net consisting of three rectangles and two right triangles. We are given the dimensions of the rectangles (length = 8 inches, width = 4 inches) and the dimensions of the right triangles (base = 3 inches, slant height = 5 inches). The task likely involves calculating the surface area or volume of the three-dimensional shape formed when the net is folded.
Answer
The total surface area is $108$ square inches.
Answer for screen readers
The total surface area of the triangular prism is $108$ square inches.
Steps to Solve
- Identify the 3D shape
The 2D net described will form a triangular prism when folded. The rectangles form the rectangular faces, and the two right triangles form the triangular bases.
- Calculate the height of the right triangle
We're given the base (3 inches) and hypotenuse (5 inches) of the right triangle. We need to find the height to calculate the area of the triangle. Using the Pythagorean theorem: $a^2 + b^2 = c^2$ where $a$ is the base, $b$ is the height, and $c$ is the hypotenuse. $3^2 + b^2 = 5^2$ $9 + b^2 = 25$ $b^2 = 16$ $b = 4$ inches
- Calculate the area of one triangle
The area of a triangle is given by: $Area = \frac{1}{2} \cdot base \cdot height$ $Area = \frac{1}{2} \cdot 3 \cdot 4 = 6$ square inches
- Calculate the total area of triangles
Since there are two triangles: $TotalArea_{triangles} = 2 \cdot 6 = 12$ square inches
- Calculate the area of one rectangle
The rectangles have a length of 8 inches and a width of 4 inches. $Area = length \cdot width = 8 \cdot 4 = 32$ square inches
- Calculate the total area of rectangles
Since there are three rectangles: $TotalArea_{rectangles} = 3 \cdot 32 = 96$ square inches
- Calculate the total surface area of the triangular prism
The total surface area is the sum of the areas of the two triangles and the three rectangles: $TotalSurfaceArea = TotalArea_{triangles} + TotalArea_{rectangles} = 12 + 96 = 108$ square inches
The total surface area of the triangular prism is $108$ square inches.
More Information
A triangular prism is a three-sided prism whose bases are triangles. It is a polyhedron made of one triangular face, three rectangular faces, and two triangular bases.
Tips
A common mistake is to forget to multiply the area of a single triangle or rectangle by the number of triangles or rectangles present in the net. Another mistake could be calculating the area of the triangle wrong by not finding the correct height.
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