Unit 3: Mensuration, Calendar and Clocks - Subjective Questions
PEA308 — Advanced Analytical Skills-Ii • Practice Questions with Detailed Answers
20 questions
A running track consists of a rectangle of length and width , with a semicircle attached at each shorter end. Find the perimeter and area of the track.
Step 1: Identify the combined shape
The two semicircles together form a complete circle whose diameter is .
Perimeter:
Only the two longer sides of the rectangle and the circumference of the complete circle are included.
Using :
Area:
Using :
Answer: The perimeter is and the area is .
Compare the areas of a square and a circle having the same perimeter . Which figure encloses the greater area?
Area of the square:
If the side of the square is , then
Therefore,
Area of the circle:
If the radius is , then
Therefore,
Comparison:
Since , the circle has the greater area.
Conclusion: Among a square and a circle with the same perimeter, the circle encloses more area.
A rectangular garden is long and wide. A path of uniform width is constructed inside the garden along its boundary. Find the area of the path and the cost of paving it at per square metre.
Outer dimensions:
Since the path is wide on every side, the dimensions of the inner rectangle are
Area of the path:
Cost of paving:
Answer: The area of the path is , and the paving cost is .
Define the lateral surface area, total surface area and volume of a cube. Hence calculate them for a cube of side .
For a cube of side :
- Lateral surface area: Area of its four side faces.
- Total surface area: Area of all six faces.
- Volume: Space occupied by the cube.
For :
Answer: The lateral surface area is , total surface area is , and volume is .
A solid cuboid measuring is melted and recast into cubes of side . Find the number of cubes formed and compare the total surface area before and after recasting.
Volume of the cuboid:
Volume of one cube:
Number of cubes:
Original total surface area:
For a cuboid,
Total surface area after recasting:
Surface area of one cube is
For cubes:
Increase in surface area:
Answer: cubes are formed. The total surface area increases from to , an increase of .
State and explain the formulas for the surface area and volume of a sphere. Find these quantities for a sphere of radius .
For a sphere of radius :
- Surface area:
- Volume:
For :
Using :
The volume is
Using :
Answer: The surface area is , and the volume is .
A hemispherical bowl has an internal radius of . Find its capacity and the area of its inner curved surface.
For a hemisphere of radius :
- Capacity or volume:
- Curved surface area:
Given :
Since :
The inner curved surface area is
Answer: The capacity is , or , and the inner curved surface area is .
Derive the formula for the curved surface area of a right circular cone and calculate the curved and total surface areas of a cone having radius and height .
Derivation:
When the curved surface of a cone is opened, it forms a sector of a circle of radius equal to the slant height .
The arc length of the sector equals the circumference of the cone's base:
The area of a sector can be written as
Therefore,
For and , the slant height is
Curved surface area:
Total surface area:
Answer: The curved surface area is , and the total surface area is .
A hollow cylindrical pipe has external radius , internal radius and length . Find the volume of material used and the total surface area of the pipe.
Let the external radius be , internal radius be and length be .
Volume of material:
Using :
Total surface area:
It includes the outer curved surface, inner curved surface and two annular ends.
Using :
Answer: The volume of material is , and the total surface area is .
A toy is formed by mounting a cone on a cylinder of the same radius. The common radius is , the cylinder's height is and the cone's height is . Find the total volume and exposed surface area of the toy.
The radius is . The cylinder has height , and the cone has height .
Slant height of the cone:
Total volume:
Exposed surface area:
The common circular face is internal and is not included. The exposed area consists of the cylinder's curved surface, the cone's curved surface and the cylinder's lower base.
Answer: The total volume is , and the exposed surface area is .
Explain the concepts of an ordinary year, a leap year and odd days. State the leap-year rule for century and non-century years.
- An ordinary year has days.
- A leap year has days because February contains days instead of .
- An odd day is a day remaining after complete weeks have been counted.
Since
an ordinary year has one odd day.
Since
a leap year has two odd days.
Leap-year rules:
- A non-century year is a leap year if it is divisible by .
- A century year is a leap year only if it is divisible by .
Thus, and are leap years, while and are not leap years.
Odd days are useful because the day of the week advances by one day after an ordinary year and by two days after a leap year.
Using the odd-days method, determine the day of the week on August .
Step 1: Count completed years
Before , there were completed years.
Number of leap years among them:
Total odd days contributed by the completed years:
Therefore, taking January of year as Monday, January was Wednesday.
Step 2: Count days from January to August
Days completed before August:
Moving two days forward from Wednesday gives Friday.
Answer: August was a Friday.
If January was a Wednesday, determine the date and day exactly days later.
The year is a leap year, so February has days.
Find the date:
January is the th day of the year. Therefore, days later is the
Cumulative days at the end of each relevant month are:
- January:
- February:
- March:
Thus, the th day falls in April:
So the date is April .
Find the day:
Two days after Wednesday is Friday.
Answer: Exactly days after January was Friday, April .
Explain when two years have identical calendars. Hence find the next year after that has the same calendar as .
Two years have identical calendars when:
- Both are of the same type, meaning both are ordinary years or both are leap years.
- Both begin on the same day of the week.
The first day of the next year shifts by:
- One day after an ordinary year.
- Two days after a leap year.
The year is an ordinary year. Track the shifts from the beginning of :
- After : total shift
- After : total shift
- After : total shift
- After leap year : total shift
- After : total shift
- After : total shift
Therefore, begins on the same weekday as . Also, both and are ordinary years.
Answer: The next year with the same calendar as is .
How many odd days are there in , , and years? Explain why the Gregorian calendar repeats after years.
For years:
There are ordinary years and leap years.
For years:
For years:
For years:
In years, the number of leap years is
Thus, there are ordinary years and leap years.
Therefore:
- years have odd days.
- years have odd days.
- years have odd day.
- years have odd days.
Conclusion: A -year Gregorian cycle contains a whole number of weeks and preserves the leap-year pattern. Hence, the Gregorian calendar repeats after years.
Explain how the hour and minute hands of a clock move. Derive the formula for the angle between them at hours and minutes.
Minute hand:
The minute hand covers in minutes. Therefore, its speed is
After minutes, its angular position from is
Hour hand:
The hour hand covers in hours, so it moves
It also moves per minute. Thus, at hours and minutes, its position is
Therefore, the angle between the hands is
If this value exceeds , the smaller angle is
This formula is the basis of angle-related clock problems.
Calculate the smaller angle between the hands of a clock at .
At , and .
Position of the hour hand:
Position of the minute hand:
Difference between the positions:
Since , it is already the smaller angle.
Answer: The smaller angle between the hands at is .
At what exact time between and o'clock do the hour and minute hands coincide?
Let the hands coincide minutes after o'clock.
At , the hour hand is at
After minutes:
- Position of the hour hand:
- Position of the minute hand:
For coincidence:
The fraction minute equals
Answer: The hands coincide at
approximately.
How many times do the hands of a clock coincide and form a right angle in hours? Explain your answer.
Coincidences:
The minute hand gains on the hour hand at the relative speed
During hours, the relative angular gain is
The number of complete relative revolutions is
Hence, the hands coincide times in hours.
Right angles:
During each relative revolution, the hands are at right angles twice: once at a separation of and once at .
Therefore,
Hence, the hands form a right angle times in hours.
Answer: In hours, the hands coincide times and form a right angle times.
A clock gains minutes every hours. It is set correctly at AM. What is the actual time when the clock shows PM on the same day?
The clock gains minutes in every minutes of actual time. Therefore, in actual minutes, it records
Thus,
From AM to the displayed PM, the clock records hours, or minutes.
Let the actual elapsed time be minutes. Then
Now,
Adding this to AM gives
Answer: When the fast clock shows PM, the actual time is PM.
A running track consists of a rectangle of length and width , with a semicircle attached at each shorter end. Find the perimeter and area of the track.
Step 1: Identify the combined shape
The two semicircles together form a complete circle whose diameter is .
Perimeter:
Only the two longer sides of the rectangle and the circumference of the complete circle are included.
Using :
Area:
Using :
Answer: The perimeter is and the area is .
Did this save you a night before the exam?
LPU Notes is free, and it stays free. Ads cover part of the server bill. The rest comes out of a student's own pocket: the domain, the storage, and keeping the site up through the weeks everyone needs it at once.
The payment button didn't load. An ad blocker or a filtered network is the usual reason. to try again.
Nothing here is ever locked, and nothing unlocks. Chip in only if it was worth it. What it pays for →