Chapter 11: Mensuration

Solution: The correct answer is option (c) 54

Explanation:

Given: The cube side = 5 cm

The side of cube 5cm is cut into 5 equal parts, in which each of 1 cm

Therefore, the total number of cubes of side 1 cm = 25 + 25 + 25 + 25 + 25 = 125

In one face of the cube, there is a total of 9 small cubes painted.

We know that there are 6 faces in a cube.

Thus, a total of 9 x 6 faces will have one face painted.

(i.e.), 54

Solution: The correct answer is option (c) 1: 4

Explanation:

Given: The cube side is 4cm

The side of cube 4cm is cut into small cubes, in which each of 1 cm

Therefore, the total number of cubes = 4 x 16 = 64 cubes

Thus, the number of cut-out cubes = 64/1

Now, the surface area of the cut-out cubes = c x 1 cm²

The surface area of the original cube = 6 x 4²

Hence, the required ratio obtained is:

= 6 x 4² / = 64 x 6

= 1: 4

Solution: The correct answer is option (b) 1/2 of the original square

Explanation:

Let “a” be the side of the square sheet

Thus, the area of the bigger square sheet = a² …. (1)

Now, the circle of maximum possible size from it is given as:

The radius of the circle = a/2 … (2)

Then the diameter = a

We know that any square in the circle of maximum size should have the length of the diagonal, which is equal to the diameter of the circle.

It means that the diagonal of a square formed inside a circle is “a”

Hence, the square side = a / √2

Thus, the area of square = a² / 2

By equating the equations (1) and (2), we will get:

The area of the resultant square is ½ of the original square.

Solution: The correct answer is option (a) lw /2

Explanation:

We know that the area of a triangle is (1/2) x base x height

Let ABCD be a triangle with length “l” and width “w”.

Here, we have to construct a triangle of maximum area inside the rectangle in all possible ways.

Now, the maximum base length is “l”

Maximum height is “w”.

Therefore, the area of the largest triangle is (1/2) x l x w.

Solution: The correct answer is option (b) Volume of the cylinder will remain unchanged.

Explanation:

We know that the volume of a cylinder is π × r² × h

We know that the base radius and height of the cylinder are “r” and “h”, respectively.

Now, height “h” becomes (1/4)h and “r” becomes “2r”, then the volume of the cylinder is:

(V) = π × 4r² × (1/4) h = πr²h = v

Therefore, the volume of the new cylinder = the volume of the original cylinder.

Solution: The correct answer is option (c) Curved surface area of the cylinder will be halved.

Explanation:

We know that the curved surface area of a cylinder with radius “r” and height “h” is given as

The curved surface area of a cylinder = 2πrh … (1)

Now, the new curved surface area of the cylinder with radius 2r and height (1/4)h, then the new curved surface area is

= 2π(2r)(1/4)h

= πrh

Now, multiply and divide the new curved surface area by 2, we will get

= (1/2) (2) πrh …. (2)

Now, by comparing (1) and (2), we get:

The new curved surface area of a cylinder is (1/2) times of the original curved surface area of a cylinder.

Solution: The correct answer is option (d) None of the above.

Explanation:

We know that the total surface area of a cylinder is 2π r(h + r), when the radius is “r” and the height is “h”.

If the radius is 2r and the height is (1/4)h, then the total surface area becomes,

= 2π (2r) ((1/4)h + 2r)

= 4 πr [(h+8r)/4]

= πr (h+8r)

Solution: The correct answer is option (a) 30 cm³

Explanation:

It is given that the coterminous faces of a cuboid is given as:

l × b = 6

l × h = 15

b × h = 10

The formula for volume of a cuboid is l × b × h

l² × b² × h² = 6 × 15 × 10

√ (lbh) = √ (900) = 30

Solution: The correct answer is option (b) 6r

Explanation:

We know that a hexagon contains six equilateral triangles, where one of the vertices of each equilateral triangle meet at the centre of the hexagon.

The radius of the smallest, which is inscribing the hexagon, is equal to the sides of the equilateral triangle.

Therefore, the perimeter of a regular hexagon is 6r, as each side of the hexagon is equal to the radius of the hexagon.

Solution: The correct answer is option (c) 4000

Explanation:

Given that, the dimensions of the godown are 40 m, 25 m and 10 m

Volume = 40 m × 25 m × 10 m = 10000 m³

Given that the volume of each cuboidal box is 2 m × 1.25 m × 1 m = 2.5 m³

Hence, the total number of boxes to be filled in the godown is

= 10000/2.5 = 4000