The radius of a sphere is increased by x%. The volume of the sphere increases by 72.8%. Find the value of x.
Answer: (1 + x100)3 = 1.728 = 1.23 ⇒ 1 + x100 = 1.2 ⇒ x = 20.
Step-by-step solution
Idea: New volume = (radius factor)3 × old volume. A 72.8% increase means the volume is multiplied by 1.728; take the cube root.
- Volume factor = 1 + 0.728 = 1.728, so (1 + x100)3 = 1.728.1 mark
- 1.728 = 1.2 × 1.2 × 1.2 (as 123 = 1728), so 1 + x100 = 1.2 and x = 20.1 mark
x = 20
Check: Radius × 1.2 ⇒ volume × 1.728 = increase of 72.8% ✓.
Answer to write in the exam
(1 + x100)3 = 1.728
1 + x100 = ∛1.728 = 1.2
∴ x = 20
Common mistakes that cost marks
- Dividing 72.8 by 3 and answering about 24.3.
- Taking the cube root of 0.728 instead of 1.728.
How this can come in the exam
MCQ (1 mark)
The volume of a sphere becomes 8 times. Its radius becomes
- 8 times
- 4 times
- 2 times
- √8 times
Show answer
(C) 2 times
∛8 = 2.
Try one yourself
The volume of a sphere increases by 33.1% when its radius increases by y%. Find y.
Show answer
(1 + y100)3 = 1.331 = 1.13 ⇒ y = 10.
More questions like this
- The hemispherical dome of a building needs to be painted (see the figure). If the circumference of the base of the dome is 35.2 m, find the cost of painting it, given the cost of painting is ₹ 10 per 100 cm2.
- We learned about different 3D solids. Identify and list objects or structures around you that are cubical, cuboidal, cylindrical, conical, spherical, or pyramidal in shape, or closely resemble these shapes.
- Activity: Find the approximate volumes of a gas cylinder, a bucket, any cylindrical vessel, a cupboard, a water tank that you can find near you (use 1000 cm3 = 1 litre).
- Lallan’s golgappa (a small hollow ball made of wheat) stall has a cylindrical vessel with pani (spiced water) in it. The vessel is of radius 10 cm and it has pani up to a height of 50 cm. Estimate the number of golgappas he can serve assuming all golgappas are spheres. Before calculating, make a guess.
Can you guess what number this could be — is it in the hundreds, thousands, or more? - Anirban saw a few tennis balls lying around. He wondered, “How many tennis balls can fit in an empty classroom?”. The classroom is of dimensions 30 ft × 30 ft × 15 ft. Estimate how many tennis balls of radius 3 cm fit in this classroom. Before calculating, make a guess.