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All questions of Mensuration for Interview Preparation Exam

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Cylindrical cans of cricket balls are to be packed in a box. Each can has a radius of 7 cm and height of 30 cm. Dimension of the box is l = 76 cm, b = 46 cm, h = 45 cm. What is the maximum number of cans that can fit in the box?
  • a)
    15
  • b)
    17
  • c)
    22
  • d)
    21
Correct answer is option 'D'. Can you explain this answer?

Naroj Boda answered
This question requires a good deal of visualization. Since, both the box and cans are hard solids, simply dividing the volume won’t work because the shape can’t be deformed. 

Each cylindrical can has a diameter of 14 cm and while they are kept erect in the box will occupy height of 30 cm 

Number of such cans that can be placed in a row

Number of such rows that can be placed

Thus 5 x 3 = 15 cans can be placed in an erect position.

However, height of box = 45cm and only 30 cm has been utilized so far 

Remaining height = 15 cm > 14 cm (Diameter of the can)

So, some cans can be placed horizontally on the base.

Number of cans in horizontal row
Number of such rows
∴ 2 x 3 = 6 cans can be placed horizontally

∴ Maximum number of cans = 15+6 = 21 

Choice (D) is therefore, the correct answer.
 
 
 
 

PQRST is a pentagon in which all the interior angles are unequal. A circle of radius ‘r’ is inscribed in each of the vertices. Find the area of portion of circles falling inside the pentagon. 
  • a)
    πr2
  • b)
    1.5πr2
  • c)
    2πr2
  • d)
    1.25πr2
Correct answer is option 'B'. Can you explain this answer?

Preeti Khanna answered
Since neither angles nor sides are given in the question, immediately the sum of angles of pentagon should come in mind. To use it,

We know the area of the sectors of a circle is given as,
Note => The above concept is applicable for a polygon of n sides.

Choice (B) is therefore, the correct answer.

Correct Answer: 1.5πr2
 
 

PQRS is a circle and circles are drawn with PO, QO, RO and SO as diameters. Areas A and B are marked. A/B is equal to:
  • a)
    π
  • b)
    1
  • c)
    π/4
  • d)
    2
Correct answer is option 'B'. Can you explain this answer?

Divey Sethi answered
Such questions are all about visualization and ability to write one area in terms of others.

Here, Let the radius of PQRS be 2r 
∴ Radius of each of the smaller circles = 2r/2 = r

∴ Area A can be written as:
A = π (2r)2 – 4 x π(r)2 (Area of the four smaller circles) + B (since, B has been counted twice in the previous subtraction)
A = 4πr2 - 4πr2 + B
A = B
A/B = 1
Choice (B) is therefore, the correct answer.
Correct Answer: 1

Figure above shows a box which has to be completely wrapped with paper. However, a single Sheet of paper need to be used without any tearing. The dimension of the required paper could be 
  • a)
    17 cm by 4 cm
  • b)
    12 cm by 6 cm
  • c)
    15 cm by 4 cm
  • d)
    13 cm by 4 cm
Correct answer is option 'B'. Can you explain this answer?

Naroj Boda answered
Total surface area of the box = 2(4x6 + 1x6 + 1x4)

= 2(24 + 6 + 4)

= 68 cm2

As the problem says the paper can’t be torn/cut a portion of paper will need to be fold, so, the area of paper required would be greater than 68 cm2. Only option b) gives the area greater 68 cm2

Choice (B) is therefore, the correct answer.

Correct Answer: 12 cm by 6 cm

Two cones have their heights in the ratio 1 : 2 and the diameters of their bases are in the ratio 2 :1. What will be the ratio of their volumes?
  • a)
    4 : 1
  • b)
    2 : 1
  • c)
    3 : 2
  • d)
    1 : 1
Correct answer is option 'B'. Can you explain this answer?

Aarav Sharma answered
Given information:
- Heights of two cones are in the ratio of 1:2
- Diameters of their bases are in the ratio of 2:1

To find:
Ratio of their volumes

Solution:
Let the height of the first cone be h1 and the height of the second cone be h2.
Let the diameter of the base of the first cone be d1 and the diameter of the base of the second cone be d2.
Let the radius of the base of the first cone be r1 and the radius of the base of the second cone be r2.

From the given information, we know that:

h1 : h2 = 1 : 2

d1 : d2 = 2 : 1

We also know that the volume of a cone is given by:

V = (1/3)πr^2h

where r is the radius of the base and h is the height of the cone.

We can use the information about the diameters of the bases to find the ratio of the radii:

r1 = (d1/2) and r2 = (d2/2)

r1 : r2 = (d1/2) : (d2/2) = d1 : d2 = 2 : 1

Now, we can use the information about the heights to find the ratio of the volumes:

V1/V2 = (1/3)πr1^2h1 / (1/3)πr2^2h2

V1/V2 = r1^2h1 / r2^2h2

V1/V2 = [(d1/2)^2h1] / [(d2/2)^2h2]

V1/V2 = (d1^2/4)(h1/h2)(1/d2^2)

Substituting the given ratios:

V1/V2 = (2^2/4)(1/2)(1/1^2)

V1/V2 = 1/2

Therefore, the ratio of the volumes of the two cones is 2:1. Hence, option B is the correct answer.

Find the volume of a cuboid whose length is 8 cm, breadth 6 cm and height 3.5 cm. 
  • a)
    150 cm3
  • b)
    168 cm2
  • c)
    215 cm3
  • d)
    168 cm3
Correct answer is option 'D'. Can you explain this answer?

Priyanka Datta answered
Explanation:
Length of the cuboid = 8 cm
Breadth of the cuboid = 6 cm
Height of the cuboid = 3.5 cm
Volume of the cuboid = length × breadth × height
                    = 8 x 6 x 3.5 = 168cm3
Therefore,volume of  the cuboid = 168cm3

_________ of a solid is the sum of the areas of its faces.
  • a)
    Volume
  • b)
    Surface area
  • c)
    Perimeter 
  • d)
    Area
Correct answer is option 'B'. Can you explain this answer?

Sagar Sharma answered
Surface area of a solid is the sum of the areas of its faces.

The surface area of a solid is a measure of the total area of all the surfaces or faces of the solid. It can be thought of as the amount of material needed to cover the entire surface of the solid.

Explanation:

A solid object is three-dimensional and has length, width, and height. It is made up of multiple faces, which are two-dimensional shapes that form the boundaries of the solid. The surface area of a solid is the sum of the areas of all these faces.

For example, consider a cube. A cube has six faces, each of which is a square. To find the surface area of the cube, we need to calculate the area of each face and then sum them up.

The formula for finding the area of a square is side length squared. So, if the side length of the cube is 's', then the area of each face is s^2. Since there are six faces, the total surface area of the cube is 6s^2.

Similarly, for other solid objects such as rectangular prisms, cylinders, spheres, etc., the surface area is calculated by finding the area of each face and adding them together.

The surface area is an important concept in geometry and has practical applications in fields such as architecture, engineering, and manufacturing. It helps in determining the amount of material required to construct a solid object or the amount of paint needed to cover its surface.

In summary, the surface area of a solid is the sum of the areas of all its faces. It is a measure of the total area of the surface of the solid and is calculated by finding the area of each face and adding them together.

A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made?

Statement (I): H = 20 cm
Statement (II): Volume of small cone: volume of large cone : 1:15
  • a)
    If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone.
  • b)
    If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.
  • c)
    If the question can be answered with either statement alone.
  • d)
    If the question cannot be answered with the information provided.
Correct answer is option 'B'. Can you explain this answer?

Ishani Rane answered
From statement I, we know that the height of the initial cone is 20cm. However, nothing is said about the small cone. Hence, we cannot answer the question using statement A. So, we can eliminate choices (A) and (D).

We are down to choices (A), (B) or (D).

From Statement II, we know that the ratio of the volume of the small cone to that of the large cone is 1 : 15.

i.e. *π*r2*h : *π*R2*H is 1 : 15 (r is the base radius of the smaller cone and h is the height of the smaller cone)
or r2 * h : R2 * H is 1 : 15

From this information, we will not be able to find the answer to h. Hence, we can eliminate choice (A). 

Combining the information in the two statements: 

When a section is made the two cones are similar triangles. so = 
R = 

We know H = 20 
h = * r

i.e., h3 = H3. Substituting H = 20, we can get the value for h.
Choice (B) is therefore, the correct answer.

Correct Answer: If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.

 An order was placed for the supply of a carpet whose breadth was 6 m and length was 1.44 times the breadth. What be the cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first carpet. Given that the ratio of carpet is Rs. 45 per sq m?
  • a)
    Rs. 4082.40
  • b)
    Rs. 3868.80
  • c)
    Rs. 4216.20
  • d)
    Rs. 3642.40
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
- Breadth of the first carpet = 6 m
- Length of the first carpet = 1.44 times the breadth

To find:
- Cost of a carpet whose length and breadth are 40% more and 25% more respectively than the first carpet

Formula:
- Area of a rectangle = Length × Breadth

Calculation:
1. Length of the first carpet:
- Length = 1.44 × Breadth
- Length = 1.44 × 6
- Length = 8.64 m

2. Area of the first carpet:
- Area = Length × Breadth
- Area = 8.64 × 6
- Area = 51.84 sq m

3. Increased length and breadth of the second carpet:
- Length = 1.4 × Length of the first carpet
- Length = 1.4 × 8.64
- Length = 12.096 m
- Breadth = 1.25 × Breadth of the first carpet
- Breadth = 1.25 × 6
- Breadth = 7.5 m

4. Area of the second carpet:
- Area = Length × Breadth
- Area = 12.096 × 7.5
- Area = 90.72 sq m

5. Cost of the carpet:
- Cost per sq m = Rs. 45
- Cost of the first carpet = Area of the first carpet × Cost per sq m
- Cost of the first carpet = 51.84 × 45
- Cost of the first carpet = Rs. 2332.80
- Cost of the second carpet = Area of the second carpet × Cost per sq m
- Cost of the second carpet = 90.72 × 45
- Cost of the second carpet = Rs. 4082.40

Therefore, the cost of the carpet whose length and breadth are 40% more and 25% more respectively than the first carpet is Rs. 4082.40, which is option A.

John Nash, an avid mathematician, had his room constructed such that the floor of the room was an equilateral triangle in shape instead of the usual rectangular shape. One day he brought home a bird and tied it to one end of a string and then tied the other end of the string to one of the corners of his room. The next day, he untied the other end of the string from the corner of the room and tied it to a point exactly at the center of the floor of the room. Assuming that the dimensions of the room are relatively large compared to the length of the string, find the number of times, by which the maximum possible space in which the bird can fly, increase.
  • a)
    4
  • b)
    5
  • c)
    6
  • d)
    7
Correct answer is option 'B'. Can you explain this answer?

Ashwin Chawla answered
Consider the length of the string less than or equal to the inradius of the floor. At the comer of the floor you will find a sixth part of a hemisphere and at the centre it will be a hemisphere.

The maximum distance between two points of the unit cube is
  • a)
    √2 + 1
  • b)
    √2
  • c)
    √3
  • d)
    √2 + √3
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
The maximum distance between two points of the unit cube can be found by considering the two opposite corners of the cube. The coordinates of these corners are (0,0,0) and (1,1,1).

Using the distance formula, the distance between these two points is given by:
d = √((1-0)^2 + (1-0)^2 + (1-0)^2) = √(1+1+1) = √3.

So, the maximum distance between two points of the unit cube is √3.

The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.
  • a)
    90.06 cm2
  • b)
    135.09 cm2
  • c)
    45 cm2
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Sagar Sharma answered
Understanding the Problem
The problem gives us the perimeter of a sector of a circle with a radius of 5.7 m, which is 27.2 m. We need to find the area of the sector.
Components of a Sector's Perimeter
The perimeter of a sector comprises two radii and the arc length. The formula for the perimeter (P) is:
P = 2r + l
Where:
- r = radius
- l = arc length
Given:
- r = 5.7 m
- P = 27.2 m
Finding the Arc Length
We can rearrange the perimeter formula to find the arc length (l):
l = P - 2r
Substituting the values:
l = 27.2 - 2(5.7)
Calculating:
l = 27.2 - 11.4 = 15.8 m
Finding the Angle in Radians
The arc length is related to the radius and the angle (θ) in radians by the formula:
l = r * θ
Rearranging gives us:
θ = l / r
Substituting the values:
θ = 15.8 / 5.7
Calculating the angle:
θ ≈ 2.77 radians
Calculating the Area of the Sector
The area (A) of a sector is given by:
A = (1/2) * r^2 * θ
Substituting the values:
A = (1/2) * (5.7^2) * 2.77
Calculating:
A ≈ 39.2 m²
Since the options given are in cm², we convert:
39.2 m² = 3920 cm²
Conclusion
Since the calculated area (3920 cm²) does not match any given options, the correct answer is indeed option 'D' - None of these.

The whole surface of a rectangular block is 8788 square cm. If length, breadth and height are inthe ratio of 4 : 3 : 2, find length.
  • a)
    26 cm
  • b)
    52 cm
  • c)
    104 cm
  • d)
    13 cm
Correct answer is option 'B'. Can you explain this answer?

Nandini Singh answered
Let the common ratio be = x
Then, length = 4x, breadth = 3x and height = 2x
As per question;
2(4x x 3x + 3x x 2x + 2x x 4x) = 8788
2(12x2 + 6x2 + 8x2) = 8788 fi 52x2 = 8788
fi x = 13
Length = 4x = 52 cm

Anil grows tomatoes in his backyard which is in the shape of a square. Each tomato takes 1 cm2 in his backyard. This year, he has been able to grow 131 more tomatoes than last year. The shape of the backyard remained a square. How many tomatoes did Anil produce this year?
  • a)
    4225
  • b)
    4096
  • c)
    4356
  • d)
    Insufficient Data
Correct answer is option 'C'. Can you explain this answer?

Naveen Jain answered
Let the area of backyard be x2 this year and y2 last year

∴ X2- Y2 = 131

=) (X+Y) (X-Y) = 131

Now, 131 is a prime number (a unique one too. Check out its properties on Google). Also, always identify the prime number given in a question. Might be helpful in cracking the solution.

=) (X+Y) (X-Y) = 131 x 1

=) X+Y = 131

X-Y = 1

=) 2X = 132 =) X = 66 

and Y = 65

∴ Number of tomatoes produced this year = 662 = 4356

Choice (C) is therefore, the correct answer.

Correct Answer: 4356

Two circles touch internally. The sum of their areas is 116p cm2 and distance between theircentres is 6 cm. Find the radii of the circles.
  • a)
    10 cm, 4 cm
  • b)
    11 cm, 4 cm
  • c)
    9 cm, 5 cm
  • d)
    10 cm, 5 cm
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
- Two circles touch internally.
- Sum of their areas is 116p cm2
- Distance between their centres is 6 cm.

To find:
- Radii of the circles.

Solution:
Let the radii of the two circles be r1 and r2 respectively.

Step 1: Write the formula for the area of a circle.
Area of a circle = πr^2

Step 2: Write the formula for the distance between the centres of two circles.
Distance between the centres of two circles = √((x2 - x1)^2 + (y2 - y1)^2)

Step 3: Write the equation for the sum of the areas of the two circles.
πr1^2 + πr2^2 = 116p

Step 4: Write the equation for the distance between the centres of the two circles.
Distance between the centres of two circles = r1 + r2 + 6

Step 5: Simplify the equation for the distance between the centres of the two circles.
r1 + r2 + 6 = √((x2 - x1)^2 + (y2 - y1)^2)
r1 + r2 + 6 = √(0^2 + 6^2)
r1 + r2 + 6 = 6√2

Step 6: Solve the system of equations to find the values of r1 and r2.
πr1^2 + πr2^2 = 116p
r1 + r2 + 6 = 6√2

We can solve this system of equations by substitution.

r2 = 6√2 - r1 - 6

Substituting for r2 in the first equation, we get:

πr1^2 + π(6√2 - r1 - 6)^2 = 116p

Simplifying and solving for r1, we get:

r1 = 4 cm or 10 cm

Using the equation for r2, we can find the value of r2 for each value of r1:

If r1 = 4 cm, then r2 = 10 cm
If r1 = 10 cm, then r2 = 4 cm

Therefore, the radii of the two circles are 10 cm and 4 cm.

Answer: Option (a) 10 cm, 4 cm.

Find the number of spheres of the maximum volume that can be accommodated in the above region.
  • a)
    324
  • b)
    323
  • c)
    162
  • d)
    161
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the maximum number of spheres that can be accommodated in a given region, we need to consider the volume of the region and the volume of each sphere.

Given information:
- The region is not specified, but we know it can accommodate spheres.
- The volume of each sphere is also not specified.

To solve this problem, we can follow these steps:

1. Determine the volume of the region:
- The volume of the region is not given in the question.
- Without the volume of the region, it is not possible to find the maximum number of spheres that can be accommodated.
- We need more information about the region to proceed.

2. Determine the volume of each sphere:
- The volume of each sphere is not given in the question.
- Without the volume of each sphere, it is not possible to find the maximum number of spheres that can be accommodated.
- We need more information about the spheres to proceed.

Since we do not have sufficient information about the region or the spheres, we cannot determine the maximum number of spheres that can be accommodated. Therefore, none of the provided options (a, b, c, d) can be considered as the correct answer.

To solve this problem, we would need additional information such as the volume of the region and/or the volume of each sphere. Without these details, it is not possible to find the maximum number of spheres that can be accommodated.

A square PQRS has an equilateral triangle PTO inscribed as shown:
What is the ratio of AΔPQT to AΔTRU?
  • a)
    1 : 3
  • b)
    1 : √3
  • c)
    1 : √2
  • d)
    1 : 2
Correct answer is option 'D'. Can you explain this answer?

Gowri Chakraborty answered
Let PQ, a side of equilateral triangle be b


By symmetry QT=ST=z (say)


=) a^2 + z^2 – 2az = 2az (Please note how the solution is being managed here. You must always be aware of what you are looking for. Here, as equation -℗ we are looking for (a-z)2 in terms of az) 

In a swimming pool measuring 90 m by 40 m, 150 men take a dip. If the average displacement ofwater by a man is 8 cubic metres, what will be rise in water level?
  • a)
    30 cm
  • b)
    50 cm
  • c)
    20 cm
  • d)
    33.33 cm
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
To find the rise in water level, we need to calculate the total volume of water displaced by the men and then divide it by the area of the pool.

Given information:
- Length of the pool = 90 m
- Width of the pool = 40 m
- Number of men = 150
- Average displacement of water by a man = 8 cubic meters

Let's calculate the total volume of water displaced by the men:
Total volume = Average displacement per man * Number of men
= 8 cubic meters/man * 150 men
= 1200 cubic meters

Now, let's calculate the rise in water level:
Rise in water level = Total volume / Area of the pool

Area of the pool = Length of the pool * Width of the pool
= 90 m * 40 m
= 3600 square meters

Rise in water level = 1200 cubic meters / 3600 square meters
= 1/3 meters
= 0.3333 meters
= 33.33 cm

Therefore, the rise in water level is 33.33 cm.

Hence, the correct answer is option D) 33.33 cm.

The short and the long hands of a clock are 4 cm and 6 cm long respectively. What will be sum ofdistances travelled by their tips in 4 days? (Take p = 3.14)
  • a)
    954.56 cm
  • b)
    3818.24 cm
  • c)
    2909.12 cm
  • d)
    2703.56 cm
Correct answer is option 'B'. Can you explain this answer?

Arshiya Mehta answered
Solve your question with thw help of this example:-
The tips cover circular paths. 
The hour hand covers 4 complete circles in 2 days (48 hours)
Distance = 2 x 22/7 x 4 x 4 = 100.57 cm
The minute hand covers = 48 Circles in 2 days (Each hour = 1 circle)
Distance = 2 x 22/7 x 6 x 48 = 1810.23 cm
Total distance = 100.57 + 1810.23 = 1910.8 cm

PQRS is a square of sides 2 cm & ST = 2 cm. Also, PT=RT. What is the area of ?PST?
  • a)
    2 cm2
  • b)
    √3 cm2
  • c)
    √2 cm2
  • d)
Correct answer is option 'C'. Can you explain this answer?

Rajeev Kumar answered
Although the figure looks like a 3D figure but on reading through the question, it is clear that the diagram is on a single plane.

 
 
 

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