Geometry

Solids

Medium

Three spherical balls of radius 2 cm, 4 cm, and 6 cm are melted to form a new spherical ball. In this process, there is a loss of 25% of the material. What is the radius (in cm) of the new ball?

✅ Correct Option: 1

Volume of a sphere $= \dfrac{4}{3}\pi r^3$ where $r$ is the radius.

For the three original balls:

Ball 1 with radius $= 2$ cm:

$V_1 = \dfrac{4}{3}\pi(2)^3$

$V_1 = \dfrac{4}{3}\pi(8)$

$V_1 = \dfrac{32\pi}{3}$ cm³

Ball 2 with radius $= 4$ cm:

$V_2 = \dfrac{4}{3}\pi(4)^3$

$V_2 = \dfrac{4}{3}\pi(64)$

$V_2 = \dfrac{256\pi}{3}$ cm³

Ball 3 with radius $= 6$ cm:

$V_3 = \dfrac{4}{3}\pi(6)^3$

$V_3 = \dfrac{4}{3}\pi(216)$

$V_3 = \dfrac{864\pi}{3}$ cm³

Total volume of all three balls:

Total Volume $= \dfrac{32\pi}{3} + \dfrac{256\pi}{3} + \dfrac{864\pi}{3}$

Total Volume $= \dfrac{(32 + 256 + 864)\pi}{3}$

Total Volume $= \dfrac{1152\pi}{3}$

Total Volume $= 384\pi$ cm³

Since there is a 25% loss of material, 75% of the material remains.

Volume of new ball $= 75\%$ of $384\pi$

Volume of new ball $= 0.75 \times 384\pi$

Volume of new ball $= 288\pi$ cm³

Let $r$ be the radius of the new ball.

$\dfrac{4}{3}\pi r^3 = 288\pi$

$\dfrac{4}{3}r^3 = 288$

$r^3 = 288 \times \dfrac{3}{4}$

$r^3 = 216$

$r = \sqrt[3]{216}$

$r = 6$ cm

Therefore, the radius of the new ball is 6 cm.

2025: 13 May Shift 1

The base diameter of a cylinder is 21 cm and the height is 28 cm, then:

(A) Radius of cylinder = 10.5 cm

(B) Volume = 12936 cm³

(D) Total surface area = 2541 cm²

Which of the following is/ are correct?

Choose the correct answer from the options given below:

2026: 30 May Shift 2

Consider the surface area of the following solids.

(A) A cylinder with a diameter of base 14 cm and height 80 cm.

(B) A cone with a radius of 7 cm and slant height 10 cm.

(D) A hemisphere with a radius of 21 cm.

Arrange the above in decreasing order.

Choose the correct answer from the options given below:

2026: 21 May Shift 1

Consider the surface areas of the following solids:

(A) A cuboid of 10 cm long, 7 cm broad, and 5 cm high.

(B) A cylinder of radius 3.5 cm and height 60 cm.

(D) A sphere of radius 12 cm.

Arrange the above in decreasing order.

2023: 19 June Shift 1

A cone of height 24 cm and base radius 6 cm is made of modelling clay. A child reshapes it in the form of a sphere. The radius of the sphere is :

2022: 20 Aug Shift 1

A right angle triangle is rotated along the base. If the perpendicular, hypotenuse and base of the triangle are 8 cm, 10 cm and 6 cm respectively. Find the volume of the cone formed through rotation. ( $\pi = 3.14$ )

2025: 13 May Shift 2

Which of the following statements are true?

(A) If the diagonal of a cube is $6\sqrt{3}$ cm, then its volume and surface area are equal.

(B) If a rectangular block having dimensions 6 cm x 12 cm x 15 cm is cut into an exact number of equal cubes, then the least possible number of such cubes is 27.

Choose the correct answer from the options given below:

Three spherical balls of radius 2 cm, 4 cm, and 6 cm are melted to form a new spherical ball. In this process, there is a loss of 25% of the material. What is the radius (in cm) of the new ball?

Solution

Related questions:

Keyboard Shortcuts

CUET General Test Past Year Questions (Topic-Wise)