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GATE 2024 EC GA Q.7: solution and answer

GATE 2024 · ECGA Q.7 · 2 marks · MCQ

Four identical cylindrical chalk-sticks, each of radius cm and length cm, are bound tightly together using a duct tape as shown in the following figure.

The width of the duct tape is equal to the length of the chalk-stick. The area (in ) of the duct tape required to wrap the bundle of chalk-sticks once, is

GATE 2024 EC GA question 7: An oblique 3-D line drawing of the taped bundle.Tap to open full size
Figure from the GATE 2024 EC paper. It is part of the question.
Figure description

An oblique 3-D line drawing of the taped bundle. The near end face, at lower left, shows four equal circles packed in a 2 x 2 square arrangement, each touching its two neighbours. The perspective makes the square look like a tilted parallelogram: circles at upper-left, upper-right, lower-left and lower-right. The tape outline runs tangent around the four circles: four straight segments joined by rounded corners, one corner around each circle. In the upper-right circle a short arrow from the centre (marked by a dot) to the rim is labelled r. From the end face the bundle runs back towards the upper right as a long box with rounded longitudinal edges; the far end at the upper right shows the rounded corners of the wrapped outline. Along the lower-right edge, a double-headed arrow between two dashed extension lines is labelled l and gives the length of the sticks.

  1. (A)

  2. (B)

  3. (C)

  4. (D)

Answer and solution

ANSWER

The tape needed is , option (D).

Official answer: DSource: Official answer key, GATE 2024 EC (IISc Bengaluru)

  1. Look at the cross-section. The four circles of radius sit in a square. Neighbouring circles touch, so their centres form a square of side . The tape is the tight band around the four circles, and its length is the perimeter of that band.

  2. Straight parts: along each side, the tape touches two neighbouring circles on the same side. It is therefore parallel to the line joining their centres and exactly as long, . There are 4 sides, so the straight length is .

  3. Curved parts: at each outer corner the tape wraps round one circle, between the tangent points of two neighbouring sides. The radii to those two points are perpendicular to their sides, and the sides are perpendicular to each other, so the arc spans , a quarter circle. Four quarter arcs make one full circle, . (Check: the turning angles of any closed convex band add up to .)

  4. Band length:

  5. Unrolled, the tape is a rectangle. Its width equals the stick length cm, and its length is because the bundle is wrapped once:

  6. This matches option (D).

KEY IDEA

A tight band around touching equal circles has two parts. The straight parts add up to the perimeter of the polygon joining the centres, here . The curved parts always add up to exactly one full circle, , because the band turns through in total. Unrolled, the tape is a rectangle: band length times stick length.

WATCH OUT FOR

Using the diameter in place of the radius in the arc term ( or counted per corner), or counting each straight side as (two diameters) instead of the centre-to-centre distance . That gives and leads to option (C). Using as the width gives options (A) and (B).

From the GATE 2024 EC paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

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