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(a)
The sketch and label of a rectangular solid and the pyramid formed by the angular surface edges.
Answer to Problem 12A
The sketch and label of a rectangular solid and the pyramid formed by the angular surface edges is
Explanation of Solution
Consider the angle ∠A=23°10′ of triangle ADE, the angle ∠B=35°00′ of triangle BDE.
There are mainly three types of views such as top view, front view and right side view. The rectangular solid and pyramid is formed using the trihedral method. In this method all the planes are perpendicular to each other and eight right triangles are formed. The object is placed in these right angles to take the projection.
The steps of the sketching of the rectangular solid and the pyramid formed by the surface to be cut and the extended sides of the block are following:
1.Draw a rectangular solid surface block.
2.Draw triangles on the extended surface of the rectangular solid surface block.
3.Cut the extended surface of the rectangular solid surface block as per the drawn triangle on the surface of the block to form a pyramid on the extended sides of the block.
4.The sketch of a pyramid formed on surface of the extended sides of the rectangular solid surface block.
5.Identify the angle ∠R and ∠C and the given angles in the sketch of a pyramid formed on surface of the extended sides of the rectangular solid surface block.
Figure-(1)
(b)
The angle ∠R.
Answer to Problem 12A
The angle ∠R is 31.37°.
Explanation of Solution
Write the expression for the angle ∠B using the triangle BDE.
tanB=DEBD ...... (I)
Here, the length of the side DE is DE and the length of the side BD is BD.
Write the expression for the angle ∠A using triangle ADE.
tanA=DEAD ...... (II)
Here, the length of the side AD is AD.
Write the expression for the angle ∠R using triangle CBD.
tanR=BDBC ...... (III)
Here, the length of the side AD is AD.
Calculation:
Consider the length of the side DE=1 unit.
Substitute 1 unit for DE and 35°00′ for B in Equation (I).
tan35°00′=1 unitBDBD=1 unittan35°00′BD=1 unit0.70020BD=1.4281 unit
Substitute 1 unit for DE and 23°10′ for A in Equation (II).
tan23°10′=1 unitADAD=1 unittan23°10′AD=1 unit0.4271AD=2.3413 unit
The length of the side AD and BC are same. Therefore, the length of the side BC is 2.3413 unit.
Substitute 2.3413 unit for BC and 1.4281 unit for BD in Equation (III).
tanR=1.4281 unit2.3413 unittanR=0.6099R=tan−1(0.6099)R=31.37°
Conclusion:
The angle ∠R is 31.37°.
(c)
The angle ∠C.
Answer to Problem 12A
The angle ∠C is 20.16°.
Explanation of Solution
Write the expression for the angle ∠R using the triangle CBD.
sinR=BDCD ...... (IV)
Write the expression for the angle ∠C using triangle CDE.
tanC=DECD ...... (V)
Calculation:
Consider the length of the side DE=1 unit.
Substitute 1.4281 unit for BD and 31.37° for R in Equation (IV).
sin31.37°=1.4182 unitCDCD=1.4182 unitsin31.37°CD=1.4182 unit0.5205CD=2.7246 unit
Substitute 2.7246 unit for CD and 1 unit for DE in Equation (V).
tanC=1 unit2.7246 unittanC=0.3672C=tan−1(0.3672)C=20.16°
Conclusion:
The angle ∠C is 20.16°.
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Chapter 76 Solutions
Mathematics for Machine Technology
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