Question

A large ball 2 m in radius is made up of a rope of square cross section with edge length 4 mm. Neglecting the air gaps in the ball, what is the total length of the rope of the nearest order of magnitude?​

Answer 1

Solution

Volume of Rope = Volume of ball

r (radius of ball) = 2 m

l (cross-section with edge length) = 4 mm = 0.004 m

Volume of Rope = (edges)² * Length of rope

= (0.004)² * L m

....1

(cuboidal shape rope = L W H)

Volume of ball = $\dfrac{4}{3}\pi r^3$

= $\dfrac{4}{3}\pi (2)^3$

= $\dfrac{4}{3} \times 3.141 \times 8$

= $\dfrac{100.5}{3}$

= 33.51 m³

....2

Equate 1 & 2

(0.004)² * L = 106.66

L = 33.51 × (0.004)¯²

L = 2.09 × 10^6 m

Answer 2

Explanation:

Given that, a large ball 2 m in radius is made up of a rope of square cross section with edge length 4 mm.

We have to find the total length of the rope of the nearest order of magnitude?.

Assume that the length of rope is x.

Volume of ball = Volume of rope

(Volume of ball = Volume of sphere)

4/3πr³ = (edge)² × length

Substitute the known values in the above formula,

4/3 π (2)³ = (4 × 10-³)² × x

4/3 × 22/7 × 8 = 16 × 10-⁶ × x

33.52 = 16 × 10-⁶ × x

33.52/(16 × 10-⁶) = x

2.095 × 10⁶ = x

Hence, the total length of the rope is 2.095 × 10⁶ m.

OR

Convert m into km. To do this, divide the given value by 1000.

Hence, the total length of the rope is 2095 km.