Match the solid with the development method it requires.
1. Cylinder 2. Cone 3. Transition piece
P. Radial line Q. Parallel line R. Triangulation
Answer: C
Parallel equal generators call for the parallel line method, generators converging on an apex for the radial method, and a surface with neither for triangulation. Identifying which of the three applies is the first step in any development.
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Match the solid with the shape of its lateral development.
1. Right cylinder 2. Right cone 3. Frustum of a cone
P. A sector of a circle Q. A rectangle R. The region between two concentric arcs
Answer: D
Cutting the apex from a cone's sector leaves the band between two arcs, which is why the apex must be located even when it has been removed. The cylinder's parallel generators give a rectangle rather than any curved pattern.
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Put the steps of developing a truncated cone in order.
1. Strike the sector using the slant height as radius 2. Locate the apex by producing the slant sides 3. Divide the base circle and step the chords along the arc 4. Mark the true distance to the cut on each radial line
Answer: A
The apex must be found first because it is the centre of the whole construction, then the sector, then the stations round its arc, and finally the cut distances measured along each radial line. Each step depends on the one before it.
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Consider the following statements about developments.
1. Every line on a development is a true length 2. A sphere can be developed exactly by fine enough subdivision 3. A cone's sector is struck with the slant height as radius
Answer: B
Correct answer: 1 and 3 only are correct
Statements 1 and 3 are correct: a development shows true lengths throughout, and a cone’s sector is struck with the slant height. Statement 2 is the one thing subdivision cannot achieve - a sphere has double curvature and no amount of refinement makes an approximation exact, which is why zone and lune methods are described as approximations.
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One of these four surfaces cannot be developed exactly. Which is it?
Answer: C
Correct answer: a sphere
A surface can be developed exactly only if it is single curved or plane, so that it can be unrolled without stretching. Cylinders, cones and prisms all qualify, oblique or not. A sphere is doubly curved and cannot be flattened without distortion - the same fact that makes every map projection a compromise.
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A development is required for all these purposes save one. Which does not need it?
Answer: D
Correct answer: calculating the volume enclosed by the solid
A development is a pattern for the surface, so it serves anything involving cutting, marking out or forming sheet material. Volume is a property of the enclosed space and is computed from the dimensions directly, with no pattern needed.
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A worker strikes a cone's development using the vertical height as radius. The pattern will
Answer: A
Correct answer: be too small to close round the base circle
The vertical height is shorter than the slant height, so the arc is struck at too small a radius and the sector formed cannot reach from apex to base rim. Rolled up it gives a cone of the wrong proportions that will not meet the base circle - which is why the true length must be taken from the elevation rather than assumed.
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A transition piece pattern made by the radial line method proves unusable. The reason is that the surface has
Answer: B
Correct answer: no single apex on which the generators converge
The radial line method fans the pattern about an apex, which exists only where the generators converge to a point. A transition piece joining, say, a square to a circle has no such apex, so it is developed by triangulation instead - dividing the surface into triangles whose true lengths are found individually.
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Assertion: the apex of a complete cone must be located to develop its frustum. Reason: the two arcs bounding the pattern are concentric about that apex.
Answer: C
Correct answer: both the assertion and the reason are true and the reason explains the assertion
Both are true and the reason gives the mechanism. The frustum’s pattern is the region between two arcs, and both are struck from the apex of the complete cone - so that point must be located, by extending the sloping sides in the elevation until they meet, before the pattern can be drawn at all.
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Assertion: a development may contain foreshortened lines provided the scale is stated. Reason: a development is a projection like any other view.
Answer: D
Correct answer: both statements are false, since a development preserves every true length
A development is not a projection: it is the surface itself unrolled, so every line on it is the true length of the corresponding line on the solid. That is precisely what makes it usable as a cutting pattern, and a foreshortened line on it would produce a piece of the wrong size.
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