Answer:
10 mm
Explanation:
You want to know the height of a triangular prism with the dimensions and angles shown.
Right triangle
The area of each right triangle base is ...
A = 1/2bh
A = 1/2(20 mm)(30 mm) = 300 mm²
Lateral area
The lateral area is the sum of the areas of the rectangular faces:
LA = (20 mm)(h) +(30 mm)(h) +(40 mm)(h) = (90 mm)h
Total area
The total surface area is the sum of the lateral area and the area of two bases:
SA = 1500 mm² = (90 mm)h +2×300 mm²
900 mm² = (90 mm)h . . . . . . . . . . . . . . . . . . . subtract 600 mm²
10 mm = h . . . . . . . . . . . . . . . . . . divide by 90 mm
The height of the prism is 10 mm.
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Additional comment
A right triangle with sides 20 mm and 30 mm will have a hypotenuse of length about 36.06 mm, not 40 mm. The answer to this question will be different, depending on which dimensions you take to be true.
If you take the side dimensions to be true, the largest angle in the triangle is about 104°, not 90°, and its area is about 290 mm², not 300 mm².
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