Unit 3: Mensuration, Calendar, and Clocks - Subjective Questions

Definition: A cube is a three-dimensional solid having six congruent square faces, twelve equal edges, and eight vertices.

• Area of one square face .
• A cube has four lateral faces. Therefore, its lateral surface area is:
  
• A cube has six faces. Therefore, its total surface area is:
  
• Volume is the product of length, breadth, and height. Since all three dimensions are equal to :
  

Thus, the required formulas are LSA , TSA , and volume .

The volume of the original cube is:

The volume of one smaller cube is:

Therefore, the number of smaller cubes is:

The surface area of the original cube is:

The combined surface area of the smaller cubes is:

Therefore:

Hence, smaller cubes are formed, and their combined surface area is four times the surface area of the original cube.

Let the length, breadth, and height of a cuboid be , , and , respectively.

• The four lateral faces have total area:
  
• The cuboid has three pairs of congruent rectangular faces. Hence:
  
• Its volume is:
  
• The diagonal of the base is obtained using the Pythagorean theorem:
  
• Applying the Pythagorean theorem again, the space diagonal is:
  

These formulas are used to calculate the covering area, capacity, and longest internal distance of a cuboid.

Step 1: Calculate the capacity

The volume of the tank is:

Since litres:

Step 2: Calculate the inner surface area

Since the tank is open, the painted area includes the base and four walls:

Step 3: Calculate the painting cost

Therefore, the tank has a capacity of litres, and the cost of painting its inner surface is .

A sphere is the set of all points in three-dimensional space that are at a constant distance from a fixed point called its centre.

The surface area of a sphere is four times the area of a circle having the same radius:

The volume can be obtained by integration. Consider a sphere represented by:

A circular cross-section at position has area:

Therefore:

A sphere has only one continuous curved surface. It has no flat bases or separate lateral faces. Hence, its curved surface area and total surface area are both:

Given radius:

The surface area of the sphere is:

The volume of the sphere is:

Therefore, the surface area is , and the volume is .

A hemisphere is one-half of a sphere formed by cutting the sphere through its centre.

• The curved surface area of a sphere is . Therefore, the curved surface area of a hemisphere is:
  
• A hemisphere also has a circular base of area . Thus, its total surface area is:
  
• The volume of a sphere is . Therefore, the volume of a hemisphere is:
  

The curved surface area excludes the circular base, whereas the total surface area includes it.

Given:

The capacity is equal to the volume of the hemisphere:

Since mL, the capacity is:

The inner curved surface area is:

Therefore, the bowl has a capacity of mL and an inner curved surface area of .

For a right circular cone, the radius, height, and slant height form a right triangle. Therefore:

When the curved surface is opened, it forms a sector of a circle of radius . Its arc length equals the circumference of the cone's base, . Comparing the sector with a complete circle of radius gives:

The total surface area includes the circular base:

A cone occupies one-third of the volume of a cylinder with the same base and height. Hence:

Thus, the required formulas are CSA , TSA , and volume .

Given:

The slant height is:

The curved surface area is:

The total surface area is:

The volume is:

Therefore, the slant height is cm, CSA is , TSA is , and volume is .

Let a right circular cylinder have radius and height .

If its curved surface is cut vertically and opened, it forms a rectangle:

• Length of the rectangle circumference of the base .
• Breadth of the rectangle .

Therefore, the curved surface area is:

The cylinder has two circular bases, each of area . Thus:

The volume equals base area multiplied by height:

Hence, CSA , TSA , and volume .

Let the external radius be cm, the internal radius be cm, and the length be cm.

Volume of material:

Total surface area:

The external curved surface area is:

The internal curved surface area is:

The area of the two annular ends is:

Therefore:

Thus, the volume of material is , and the total surface area is .

The volume of the cone is:

The volume of the sphere is:

The volume of the cylinder is:

Therefore, the ratio is:

Multiplying every term by gives:

Hence, their volumes are in the ratio:

Thus, the sphere has twice the volume of the cone, while the cylinder has three times the volume of the cone.

A calendar arranges days into weeks, months, and years.

• An ordinary year has days. Since:
  
  it contains complete weeks and odd day.
• A leap year has days. Since:
  
  it contains complete weeks and odd days.
• An odd day is the number of days left after grouping the total number of days into complete weeks.
• A year is a leap year if it is divisible by .
• A century year must be divisible by to be a leap year.

For example, is a leap year because it is divisible by . The year is not a leap year because it is not divisible by , whereas is a leap year because it is divisible by .

Use January, CE as Monday and count the number of complete days before the required date.

Step 1: Count days in complete years

There are complete years before . The number of leap years is:

Hence, the total odd-day contribution is:

Step 2: Count days before August

Days from January to July in are:

There are completed days before August. Therefore:

Step 3: Find the total shift

Four days after Monday is Friday. Therefore, August was a Friday.

Take January, CE as Monday.

There are complete years before . The number of leap years among them is:

The total number of ordinary-year shifts and leap-day shifts is:

There are completed days before January. Thus, the total number of odd days is:

Starting from Monday, a shift of three days gives Thursday:

• Monday Tuesday
• Monday Wednesday
• Monday Thursday

Therefore, January was a Thursday.

Two years have identical calendars when:

• Both years are of the same type, meaning both are ordinary years or both are leap years.
• The first day of both years falls on the same day of the week.

An ordinary year shifts the first day of the following year by day, while a leap year shifts it by days.

Starting after :

• is ordinary: cumulative shift .
• is ordinary: cumulative shift .
• is ordinary: cumulative shift .
• is leap: cumulative shift .
• is ordinary: cumulative shift .
• is ordinary: cumulative shift .

Therefore, January falls on the same weekday as January . Both and are ordinary years.

Hence, the next year having the same calendar as is .

A clock face represents and is divided into equal hour divisions and equal minute divisions.

• Angle between consecutive hour marks:
  
• Angle between consecutive minute marks:
  
• The minute hand covers in minutes. Its speed is:
  
• The hour hand covers in minutes. Its speed is:
  
• The relative speed of the minute hand with respect to the hour hand is:
  

At hours and minutes, the positions measured from are:

Therefore, the angle between the hands is:

If this value is greater than , the smaller angle is .

At , take and .

The position of the hour hand from is:

The position of the minute hand from is:

The angle between the two hands is:

Since , it is already the smaller angle.

Therefore, the smaller angle between the hands at is:

Let the required time be minutes after o'clock.

At , the hour hand is at:

After minutes:

1. Time of coincidence

For coincidence, the angular positions are equal:

Thus, the hands coincide at:

2. Time when the hands are opposite

For opposite directions, their angular difference is :

Thus, the hands are opposite at:

Therefore, between and , the hands coincide at and are opposite at .