Question

Case study based questions
An Umbrella frame is made with steel wire. It has 8 spokes which are equally spaced. If these spokes are expanded fully they are in the form of a circle with diameter of 4 meters. Steel wire is used in making the ribs and the outer frame of the umbrella to which these ribs are attached.

1)The total length of the Steel wire used is
a) 08.571 m
b) 11.571 m
c) 18.571
d) 28.571 m

2)The area between the three consecutive spokes of the umbrella
a) pi m²
b) 2 pi m²
c) 3 pi m²
d) 4 pi m²

3) The angle between any two spoke of the umbrella is
a) 30°
b/45°
c) 60°
d) 75

4) How much water proof cloth was used in making this umbrella.
a) 2 pi m2
b) 3 pi m²
c) 4 pi m2
d) 5 pi m²​

Answer 1

Answer:

1. d) 28.571 m

2. a) π m²

3. b) 45°

4. 4π m²

Step-by-step explanation:

1. Number of spokes = 8

Diameter = 4 m

So, radius = 2 m

Length of wire used in making spokes = Radius × Number of spokes

= 2 × 8

= 16 m

Circumference = 2πr = πd

= 22/7 × 4

= 12.571 m (approx.)

Total length of wire = Length of wire used in making spokes × Circumference

= 16 m + 12.571 m

= d) 28.571 m

2. Angle of the circle = 360°

Angle of each sector = 360°/Number of spokes

= 360°/8 = 45° ——————————— 1

Area of 2 sectors = 2 (θ/360 × πr²)

= 2 (45/360 × π × 2 × 2)

= 2 × 1/8 × π × 4

= 2 × 1/2 × π

= a) π m²

3. b) 45° (from equation 1)

4. Area of circle = πr² = π × 2 × 2

= c) 4π m²

Answer 2

Given :- Umbrella has 8 spokes which are equally spaced. If these spokes are expanded fully they are in the form of a circle with diameter of 4 meters.

Solution :-

(32)

→ Diameter of circle = 4 m

So,

→ Radius of circle = r = Diameter / 2 = 4/2 = 2 m

Now,

→ Length of each spoke = Radius of circle = 2 m

So,

→ Length of 8 spokes = 8 * 2 = 16 m

and,

→ Circumference of circle = 2πr = 2 * (22/7) * 2 = (88/7) ≈ 12.571 m

then,

→ Total length of the Steel wire used = Length of 8 spokes + Circumference of circle = 16 + 12.571 = 28.571 m (d)

(33)

→ Total spokes = 8

So,

→ Angle between each spoke = Angle at centre/Total spokes = 360°/8 = 45°

then,

→ The area between the three consecutive spokes of the umbrella = Area of two sectors = 2[(Angle at centre/360°) * πr²] = 2[(45°/360°) * 2 * π * 2] = (1/8) * 8π = π m² (a)

(34)

→ Total spokes = 8

So,

→ Angle between any two spoke of the umbrella = Angle at centre/Total spokes = 360°/8 = 45° (b)

(35)

→ Area of cloth used to make umbrella = Area of circle = πr² = π * 2 * 2 = 4π m² (c)

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