To solve the problem, we will need to determine the side lengths of the right triangular prism's net and then use it to find the surface area.
(a) Finding Side Lengths for the Net
Since no values are provided in the prompt, let’s assume the following side lengths for a right triangle as an example:
- Base = \(b\) mm
- Height = \(h\) mm
- Hypotenuse \(c\) can be calculated using the Pythagorean theorem: \(c = \sqrt{b^2 + h^2}\) mm
The side lengths of the rectangular part of the net correspond to the lengths of the sides of the triangular base and the height of the prism.
- Rectangles formed by the base of the triangle and height:
- Rectangle from base: \(b\) mm (base) × height of the prism \(H\) mm
- Rectangle from height: \(h\) mm (height) × height of the prism \(H\) mm
- Rectangle from hypotenuse: \(c\) mm (hypotenuse) × height of the prism \(H\) mm
If you provide the specific values for the base and height, I can provide the calculations for these side lengths.
(b) Finding the Surface Area of the Prism
The surface area \(A\) of a right triangular prism can be found by adding the areas of its two triangular bases and the three rectangular sides.
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Area of the triangular base: \[ A_{triangle} = \frac{1}{2} \times b \times h \]
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Area of two triangular bases: \[ A_{2 , triangles} = 2 \times A_{triangle} = 2 \times \left( \frac{1}{2} \times b \times h \right) = b \times h \]
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Area of the three rectangular sides:
- Area of rectangle from base: \[ A_{base} = b \times H \]
- Area of rectangle from height: \[ A_{height} = h \times H \]
- Area of rectangle from hypotenuse: \[ A_{hypotenuse} = c \times H \]
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Total surface area: \[ A = A_{2 , triangles} + A_{base} + A_{height} + A_{hypotenuse} \] \[ A = (b \times h) + (b \times H) + (h \times H) + (c \times H) \]
Combining everything, you get: \[ A = b \times h + (b + h + c) \times H \]
If you provide specific values for \(b\), \(h\), \(H\), I can give you the numerical result for the surface area. Otherwise, please substitute in the actual measurements you have.