CAT 2025 Lesson : Mensuration - Cuboid & Cube

The ratio of the length and breadth of a rectangular fish tank is 3 : 2. When 48 litre of water is poured into the tank, the water level raises to 20 cm. Then the breadth of the fish tank is

Solution

A rectangular tank is a cuboid. Let the length and breadth of the tank in cm be $3x$ and $2x$ respectively.

Volume of the tank =$l \times b \times h$ = 48 litres = 48000 $cm^3$

⇒ $3x \times 2x \times 20 = 48000$

⇒ $6 x^2 = 2400$

⇒ $x^2 = \sqrt{400} = 20$

Breadth of the tank = $2x$ = 40 cm

Answer: 40 cm

From each of the six sides of a cuboid of dimensions $8m \times 6m \times 5m$, a block with 1 metre thickness is sliced and removed. What is the volume of the portion removed from the cuboid?

Solution

Volume of the cuboid before removal = $8 \times 6 \times 5$= 240 $m^3$

When 1m of thickness is removed from each side, each of length, breadth and height reduce by 1m on each of the two sides they connect. Therefore, reduction in each of the dimensions is 2 m.

Volume of the resultant cube = $(8 - 2) \times (6 - 2) \times (5 - 2) = 6 \times 4 \times 3 = 72 \ m^3$

Volume of the portion removed = 240 – 72 = 168 $m^3$

Answer: 168 $m^3$

What is the total surface area (in cm$^{2}$) of the largest cube that can be fit into a sphere of radius 7 cm?

Solution

Let $a$ be the side of the square. The cube's longest diagonal will be the sphere's diameter $\therefore$ $\sqrt{3}a = 2 \times 7$ ⇒ $a = \dfrac{14}{\sqrt{3}}$ Total Surface Area of the cube = $6a^{2} = 6 \times \left( \dfrac{14}{\sqrt{3}} \right)^{2} =$ 392 cm$^{2}$

Answer: 392

A large cube is cut into 1000 identical smaller cubes. What is the ratio of the surface area of the larger cube to that of the sum of surface areas of the 1000 smaller cubes?

Solution

As the ratio is asked, we can assume a value for the side of the larger cube, say 10 cm (instead of a variable). The question does not state Total Surface Area (6$a^{2}$) or Lateral Surface Area (4$a^{2}$). However, the ratio will be unchanged as the constant on both sides will be cancelled. Let's use Lateral Surface Area here.

Volume of the larger cube $= 10^{3} = 1000$ cm$^{3}$

As the larger cube is divided into 1000 smaller cubes,
Volume of a smaller cube =$\dfrac{1000}{1000} = 1$cm$^{3}$
$\therefore$ side of a smaller cube = $\sqrt[3]{1} = 1$cm

LSA of larger cube = $4 \times 10^{2}$ = 400
LSA of 1000 smaller cubes $= 1000 \times 4 \times 1^{2} = 4000$

Ratio of LSAs = 400 : 4000 = 1 : 10

Answer: 1 : 10