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OTBA Maths SA II THEME 2 Q1) If canvas is used to make a cylindrical tent of radius of base 7 m., then what should be the volume of air inside the tent? Answer Surface area of cylindrical tent = 551 - 1m 2 = 2πrh + πr 2 Or, 550 = 2 × π × 7 × h + π × 7 × 7 Or, 550-1542×227×7=h h = 9 m Volume of cylinder = πr 2 h πr 2 h = (22/7) × 7 × 7 × 9 = 1386 m 3 Q2) The survey conducted by Sarah on the success rate of the team members of Brigadier Tripathi Team climbing the rock is represented in the form of a bar graph as shown below. Observe the bar graph and answer the following questions. a. In how many attempts do most of the team members who tried climbing the rock climbed the rock and what is their number? b. How many team members climbed the rock? c. How many team members climbed the rock in not more than two attempts? Answer

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OTBA Maths SA II THEME 2 Q1) If canvas is used to make a cylindrical tent of radius of base 7 m., then what should be the volume of air inside the tent? Answer Surface area of cylindrical tent = 551 - 1m2 = 2πrh + πr2 Or, 550 = 2 × π × 7 × h + π × 7 × 7 Or, 550-1542×227×7=h h = 9 m Volume of cylinder = πr2h πr2h = (22/7) × 7 × 7 × 9 = 1386 m3 Q2) The survey conducted by Sarah on the success rate of the team members of Brigadier Tripathi Team climbing the rock is represented in the form of a bar graph as shown below.

Observe the bar graph and answer the following questions. a. In how many attempts do most of the team members who tried climbing the rock climbed the rock and what is their number? b. How many team members climbed the rock? c. How many team members climbed the rock in not more than two attempts? Answer

From the bar graph, we can see that a. Most of the team members climbed the rock in two attempts, and the number of team members who climbed the rock in two attempts is 20. b. Number of team members who climbed the rock = (Total team members – Number of team members who did not attempt to climb) = 50 – 5 = 45 c. The number of team members who climbed the rock in not more than two attempts is 30. The team members who climbed the rock in one or two attempts come under this category Q3) If the juice contained in a cylindrical glass is poured into a hemispherical cup, then a. What is the percentage of juice that fills the cup? b. What is the percentage of juice that remains in the cylindrical glass? Answer Diameter of the hemispherical cup = 7 cm Diameter of tthe cylindrical glass = 7 cm Height of the cylindrical glass = 10.5 cm Volume of juice in the hemispherical cup =23πr3=23×π×72×72×72cm3 = 89.83 cm3 Volume of the cylindrical glass =πr2h=π×72×72×10.5 cm3=404.25 cm3 Volume of juice in the cylindrical glass = 23 × 404.25 = 269.50 cm3 Volume of juice in the hemispherical cup = 89.75 cm3 a. Percentage of juice that fills the cup =Volume of the juice. In the hemispherical cupVolume of juice in cylindrical glass×100=89.75×269.50×100=33.30 % b. Volume of the remaining juice = Volume of juice in the cylindrical glass – Volume of juice in the hemispherical cup = 269.50 – 89.75 = 179.75 cm3

Q1)

Consider the following information about the survey conducted by Sarah on the success rate of the team members of Brigadier Tripathi Team climbing the rock.

No. of attempts No. of people

1st attempt 10

2nd attempt 20

More than two attempts 15

Did not attempt at all 5

Total 50

What is the mode of the number of times the team members tried climbing the rock? Answer Mode is the most frequently occurring observation. Here, the mode of the number of attempts is 2 Q2)

If the juice contained in a hemispherical cup that is fully filled with juice is poured into a cylindrical glass, then: a. What percentage of the glass is filled with the juice?

b. What percentage of the glass is empty? Answer Diameter of the hemispherical cup = 7 cm Diameter of the cylindrical glass = 7 cm Height of the cylindrical glass = 10.5 cm Volume of juice in the hemispherical cup =23πr3=23×π×72×72×72 cm3=89.75 cm3 Volume of the cylindrical glass=πr2h=π×72×72×10.5 cm3=404.25 cm3 Volume of the cylindrical glass filled with juice = Volume of juice in the hemispherical

cup = 89.75 cm3 a. Percentage of the glass filled with juice Volume of cylindrical glass filled with juiceTotal Volume of the cylindrical glass×100=89.75404.25×100=22.20 % b. Volume of the empty cylindrical glass = Total volume of the cylindrical glass - Volume of cylindrical glass filled with juice = 404.25 – 89.75 = 314.5 cm3 Percentage of the glass that is empty =Empty Volume of cylindrical glassTotal Volume of the cylindrical glass×100=314.5404.25×100=77.8 %

Q3) Two canvasses of area 551 m2 were given to the students to make a big tent in the shape of a

cone above a cylinder. The stitching and wasting incurred while cutting amounted to 1 m2 each in both the canvasses. If the radii of canvasses are 7 m each, then what is the volume of air inside

the tent? Answer For the tent made by joining cylindrical and conical tents: Volume of air inside the tent = Volume of the cylindrical tent + Volume of the conical tent =πr2h+13πr2h' For the cylindrical tent, we need to find the height of the tent. The area of the sheet should be 551 m2. 1 m2 is wasted in stitching, so Surface area of the cylindrical tent = 551 - 1 m2 = πrh Or, 550 = 2 × π × 7 × h Or, 550×72×22×7=h h = 12.5 m Volume of the cylindrical tent = πr2h

= π × 7 × 7 × 12.5 = 1925 m3 Similarly, for the conical tent: Effective area of cone = 551 - 1 m2 = 550 m2 Area of cone = πrl r = 7 m πrl = π × 7 × l = 550 l = 25 m Let h be the height of the cone. l2 = h'2 + r2 h'2 = l2 - r2 Or, h'2 = 576 Or, h' = 24 m Volume of the conical tent = 13 πr2h' = 13 × π × 7 × 7 × 24= 1232 m3 3157 m3 Total volume of the tent = 1925 m3 + 1232 m3 = 3157 m3

Q1)

Out of the five team members of Brigadier Tripathi Team who did not attempt to climb the rock, one was selected to assist the teachers in preparing the performance chart of all the students. Use the given table to answer the question.

Gender of team members

Number of team members

Male 2

Female 3 What is the probability that the selected member is female? Answer Probability = total number of attempts in which event happenedTotal number of trials

Total number of team members who did not attempt to climb the rock = 5 Number of female members = 3 P (selected member is a female) =35

Q2) After the survey conducted by Sarah on the success rate of the team members of Brigadier

Tripathi Team who climbed the rock, she assigned groups to them.

No. of attempts No. of people Group assigned

1st attempt 10 A

2nd attempt 20 B

More than two attempts 15 C

Did not attempt at all 5 D

Total 50

Brigadier wanted to select one team member to guide the members of Sarah's team. Find the probability of selecting the team member from each group? What is the sum of all the probabilities found? Answer Solution: Probability=total number of attempts in which event happenedtotal number of trials a. Number of team members belonging to group A = 10 Total number of team members = 50 P(selecting the student from group A) =1050=0.2 b. Number of team members belonging to group B = 20 Total number of team members = 50 P(selecting the team members from group B) =2050=0.4

Q3)

What is the ratio of the volumes of air inside the hemispherical- and conical-shaped tents with

the same base area? Answer If the base area of the conical and hemispherical tent is the same, they must have the same radius, i.e., 7 m. To find the volume of the cone, we first need to find out the height of the cone. The conical tent uses one canvas of area 551 m2, in which 1 m2 is wasted in cutting and stitching. Effective area of cone = 551 - 1 m2 = 550 m2 Area of cone = πrl r = 7 m πrl = π × 7 × l = 550 l = 25 m Let h be the height of the cone. l2 = h2 + r2 h2 = l2 - r2 Or, h2 = 576 Or, h = 24 m Volume of the conical tent 13πr2h = 13 × π × 7 × 7 × 24 = 1232 m3 For the hemispherical tent: Radius = 7 m Volume of the hemispherical tent =23πr3=23×π×7×7×7=718.66 m3 Ratio of the volumes=volume of the conical tentvolume of the hemispherical tent=1232718.66=1.7:1

Q1) To accommodate four students, one of the students made a tent in the shape of a prism having

a rectangular base of length 11 m, in which the same area is given to each student for resting as it is given in the conical tent. What is the breadth of the base of the tent? Answer As the area given to each student is the same as in the conical tent, the area of the base of the new tent is equal to the area of the base of the conical tent. Area of the base of the conical tent =π×72×72 m2=772 m2 Area of the base of the new tent = Length × Breadth = 11 × Breadth =772 m2 Breadth =72=3.5m

Q2) 1 m2 convas is wasted in cutting and stitching while making a tent. The tent can accomodate

either four students or two teachers. What is the cost of the canvas that is wasted in the process if the cost of canvas is Rs 100 per m2? Answer Each tent can have four students. Total number of tents for students =604=15 Numbers of tents required to accommodate teachers =102=5 Total number of tents = 15 + 5 = 20 tents Canvas wasted in cutting and stitching a tent = 1 m2 Total canvas wasted = 20 × 1 = 20 m2 Cost of canvas that is wasted = 20 × 100 = Rs 2000

Q3)

The survey conducted by Sarah on the success rate of the team members of Brigadier Tripathi Team climbing the rock is represented in the form of a table given below.

No. of attempts No. of people

1st attempt 10

1nd attempts 20

More than two attempts 15

Did not attempt at all 5

Total 50 Represent the given data in the form of a pie chart. Answer For 1st attempt:Number of team members who climbed the rock in 1st attemptTotal number of team members=1050×360°=72° For 2nd attempt:Number of team members who climbed the rock in 2nd attemptTotal number of team members=2050×360°=144° For more than two attempts: Number of team members who climbed the rock in morethan 2 attemptTotal number of team members=1550×360°=108° For did not attempt:Number of team members who did not attemptTotal number of team members=1550×360°=36°

Q1) If a tent made from canvas is hemispherical in shape, then what is the radius of the tent? Answer Area of the canvas used = 551 m2 Effective area of the tent = 551 – 1 = 550 m2 Area of the hemisphere = 2πr2 = 550 r2=550×722×2=25×72 Or, r=572−−√m=9.3541 m

Q2) If one of the students makes a hemispherical tent with radius of base same as that of a conical

tent so that each student gets the same area for resting as it is given in the conical tent, then what is the area of the canvas required for the tent if it is assumed that 1 m2 of the convas is wasted in cutting and stitching. Answer If the areas of the bases of the tents are the same, then the radius of the base of the conical tent and that of the hemispherical tent must be the same and equal to 7 m. Surface area of the hemispherical tent = 2πr2 =2 × π × 7 × 7 = 308 m2 Area of the canvas required for the hemispherical tent = 308 + 1 = 309 m2

Q3) From the given data, answer the questions.

Number of seats in the bus

Number of buses of given type

15 3

25 1

If a student randomly sits in a bus without knowing its capacity, then a. the probability that he is seated in a 15-seater bus is b. the probability that he is seated in a 25-seater bus is Answer Probability =Total number of attempts in which event happenedTotal number of trials Total number of buses = 4 Number of 15-seater buses = 3 Number of 25-seater buses = 1 P(of sitting in a 15-seater bus ) =34 P(of sitting in a 25-seater bus ) =14

Q4)

If the total area of land available for camping is 3500 m2,, then much area is left uncovered after making all the tents? Assume that two teachers can live in one tent. Answer Each tent can have four students. Total number of tents for students =604=15 Numbers of tents required to accommodate teachers =102=5 Total number of tents = 15 + 5 = 20 Area of land covered by each tent πr2=227×72=154 m2 Total area covered by 20 tents = 20 × 154 = 3080 m2 Area left uncovered = 3500 m2 – 3080 m2 = 420 m2

Q5) How much more juice should the manager offer in order to fill a cylindrical glass to the brim? Answer Volume of the empty glass =13 of the total volume of glass =13×πr2h=13×227×72×72×10.5 cm3 = 134.75 cm3

Q1)

What is the ratio of team members who succeeded in rock climbing in more than two attempts to

those who did not attempt rock climbing at all? Answer Team members who succeeded rock climbing in more than 2 attempts = 15 Team members who did not attempt rock climbing at all = 5 Ratio =155=3 or 3 : 1

Q2)

In the survey conducted by Sarah, Brigedier gave some tasks to his team members. The members who climbed the rock in the first attempt were given four tasks, the members who climbed the rock in the second attempt were given three tasks, the members who climbed the rock in the third

attempt were given two tasks and the members who did not attempt rock climbing were given five tasks. Express this information in the form of a linear equation in two variables and draw the

graph for the same. (Take the number of attempts as x and the number of tasks assigned as y). Answer The above information can be written as: x 0 1 2 3

y 5 4 3 2 From the given information, we can write the relation between x and y as: y = 5 – x

The graph for the same is shown below.

Q3) If half of the students consumed juice in a cylindrical glass that is filled to two-thirds of its

capacity and half of the students consumed juice in a hemispherical cup, then what is the total volume of the juice consumed in litres? Answer Volume of juice consumed by the students in the cylindrical glass=30×23×227×72×72×10.5 cm3 Volume of juice consumed by the students in a hemispherical glass=30×23×227×72×72×72 cm3 Total volume of juice consumed =30×227×72×72×23×10.5+72= 10780 cm3 1000 cm3 = 1 L 10780 cm3 = 10.78 L

Q4)

Rock climbing demo was given to the students by the 5 members who have not attempted the climbing previously. However, after the demo, they were able to climb the rock. The success rate

of them is given as follows.

No. of attempts No. of

people

1st attempt 3

2nd attempt 1

More than two attempts 2

Total 5 Find out the percentages of the 50 members of Brigedier's team in (i) succeeding in the first attempt (ii) succeeding in two attempts (iii) succeeding in more than two attempts Answer Total number of members who climbed the rock in the first attempt = 13 Percentage of members who climbed the rock in the first attempt =1350×100=26 Total number of members who climbed the rock in the second attempt = 21 Percentage of members who climbed the rock in the second attempt =2150×100=42 Total number of members who climbed the rock in the third attempt = 16 Percentage of members who climbed the rock in the third attempt =1650×100=32

Q1)

What percentage of the team members attempted to climb the rock? Answer Number of those who attempted to climb the rock = 10 + 20 + 15 = 45 Percentage of students who attempted to climb the rock = =4550×10=90 %

Q2) If half of the students consume juice in a cylindrical glass that is filled to two-thirds of its

capacity and half of the students consume juice in a hemispherical cup, then what is the average volume of the juice consumed by each student? Answer

Volume of juice consumed by the students in a cylindrical glass=30×23×227×72×72×10.5 cm3 Volume of juice consumed by the students in a hemispherical glass=30×23×227×72×72×72 cm3 Total volume of juice consumed =30×227×72×72×23×10.5+72= 10780 cm3 Average volume of juice consumed =1078060cm3=179.66 cm3

Q3)

If a group of four students decides to make their tent hemispherical in shape using the canvas given to them, then how much space will be occupied by each student? Answer Area of the canvas used = 551 m2 Effective area of the tent = 551 – 1 = 550 m2 Area of the hemisphere = 2πr2 = 2 × 227 × r2 = 550 r2=550×722×2=25×72 Or, r = 9.35 m Volume of the hemispherical tent =23×πr3=23×π9.353 = 1712.65 m3 Space occupied by each student =1712.654=428.16 m3

Q4) What should be the radius of the hemispherical cup to keep the volume of juice same as that served in cylindrical glass? Answer Let the radius of the hemispherical cup be r. Volume of the hemispherical cup =23πr3 Volume of juice in the cylindrical glass =23π×72×72×10.5 Equating the volumes, we get: =23πr3=23π×72×72×10.5 Or,

r=72×72×10.51/3 Or, r=72(3)1/3

Q1)

If all tents are conical in shape and if four students live in each tent made for students and two

teachers live in each tent made for teachers, then what is total area of the canvas used in making

all the tents? Answer

Each tent can have four students, so the total number of tents for students is , 604i.e., 15.

Numbers of tents required to accommodate teachers = 102 = 5

Total number of tents = (15 + 5) = 20 Area of canvas used in making one tent = 551 m2

Total area of canvas used = 20 × 551 m2 = 11020 m2 Ads by ShowPasswordAd Options

Q2) How many cups of juice are required to fully fill two cylindrical glasses? Answer Diameter of the hemispherical cup = 7 cm Diameter of the cylindrical glass = 7 cm Height of the cylindrical glass = 10.5 cm Volume of juice in the hemispherical cup =23πr3=23×π×72×72×72cm3 = 89.75 cm3 Volume of the cylindrical glass =πr2h=π×72×72×10.5 cm3=404.25 cm3 Number of cups of juice required to fill the glass=Volume of the cylindrical glassVolume of hte hemispherical cup=404.25 cm389.72 cm3 Number of cups of juice required to fill two glasses = 2 × 4.5 = 9 cups (approximately) Q3)

If the height of a conical tent is reduced by 4 m, keeping the volume of the tent constant, then

what will be the new radius of the base? Answer Effective area of cone = 551 – 1 m2 = 550 m2 Area of cone = πrl r = 7 m πrl = π × 7 × l = 550 l = 25 m Let h be the height of the cone. l2 = h2 + r2 h2 = l2 - r2 Or, h2 = 576 Or, h = 24 m Volume of the conical tent =13πr2h =13×π×7×7×24 Let the new radius of the tent be r. Volume of the tent after decreasing the height by 4 m = 13 × π × r × r × 20 Equating the volumes, we get: 13×π×r×r×20=13×π×7×7×24 Or, r = 7.67 (approx.) Q1) What is the ratio of team members who succeeded in rock climbing in more than one attempt to those who did not attempt rock climbing at all? Answer Students who succeeded in rock climbing in more than one attempt = 20 + 15 = 35 Students who did not attempt rock climbing = 5 Ratio=355=7:1 Q2)

In a 15-seater bus, only two teachers are enough to maintain discipline. What is the probability

that there are three students of section B in the 15-seater bus? Answer The criteria for the distribution of students in all the buses is that each bus should have atleast

(i) two teachers (ii) two students from the same section Only one student is left that should be selected from six sections.

Probability =Total number of attempts in which event happenedTotal number of trials

Total number of sections = 6

Probability (of 3 students from section B)=16 Q3) If the drink is served to each student in a cup, then what is the total volume of the drink used? Answer Volume of the cup =23πr3 Or, 23×227×72×72×72 cm3 Total juice consumed by the students = 23×227×72×72×72×60 cm3 = 5390 cm3 Q4) How much juice is saved by the manager by serving in the hemispherical cup rather than in

2/3rds of the glass? Answer Total volume of juice that would be served if it is given in the cylindrical glass = 60 × 23× π × (3.5)2 × 10.5 cm3 = 16170 cm3 Total volume of juice that is served when it is given in the hemispherical cup = 60 ×23× π × (3.5)3 cm3 = 5390 cm3 Total volume of juice the manager saved = 16170 – 5390 cm3 = 10780 cm3 Q5)

The survey conducted by Sarah on the success rate of the team members of Brigadier Tripathi Team climbing the rock is represented in the form of a bar graph as shown below.

Observe the bar graph and answer the following questions:

1. How many team members climbed the rock in more than two attempts? 2. How many team members could not climb the rock in the first attempt?

3. How many team members climbed the rock in more than one attempt? Answer From the bar graph, we see that: 1. The number of team members who climbed the rock in more than two attempts is 15. 2. Number of team members who could not climb the rock in the first attempt = (Total team members – Number of team members who climbed the rock in the first attempt) = 50 – 10 = 40 3. Number of team members who climbed the rock in more than one attempt = (Total team members who climbed the rock – Number of team members who climbed rock in the first attempt) = 45 – 10 = 35 Q1) What percentage of juice is saved by the manager by serving the juice in the hemispherical cup

rather than the cylindrical glass? Answer Total volume of juice that would be served if the juice is given in the cylindrical glass = 60 × 23 × π × (3.5)2 × 10.5 cm3 = 16170 cm3 Total volume of juice that is served when the juice is given in the hemispherical cup = 60 × 23 × π × (3.5)3 cm3 = 5390 cm3 Total volume of juice saved by the manager = 16170 – 5390 cm3 = 10780 cm3 Percentage of juice saved by the manager = 1078016170 × 100 = 66.66 Q2)

Swati reported that the volume of air inside the tent is 1200 m3. Is she correct? If not, what

percentage error in the volume of air is there? Answer Swati is not correct. Effective area of cone = 551 – 1 m2 = 550 m2 Area of the cone = πrl r = 7 m πrl = π × 7 × l = 550 l = 25 m Let h be the height of the cone. l2 = h2 + r2 h2 = l2 - r2 Or, h2 = 576 Or, h = 24 m Volume of the conical tent = 13πr2h =13 × π × 7 × 7 × 24 = 1232 m3 Error in the volume reported by Swati = 1232 – 1200 = 32 m3 Percentage error in the volume reported by Swati = 321232 × 100 = 2.6 Q3)

In a group of four students, due to some reasons, each pair of students decided to make their own tent using half the area of canvas.

The tent formed by two students was conical in shape and of the half of the radius of the base of the orginal tent. Also. the area of the canvas wasted in stitching and cutting each tent is 0.5 m2. (a) Find:

(i) Volume of the small tent formed by two students. (ii) Ratio of the volume of the tent for two students to the volume of the tent for four students.

(b) Determine whether the combined volume of the two small tents is greater or less than the

volume of the original tent. Also, find the percentage increase or decrease in the volume of the tent. Answer Area of canvas for the small tent = 275.5 m2 Area of canvas wasted in cutting and stitching = 0.5 m2 Surface area of the small conical tent = 275 m2 Radius of base of the small tent = 72m (a) (i) To find the volume of the tent, we need to find the height of the tent. Area of the curved surface of a cone = πrl r is the radius of the cone with slant height l and height h. Here, πrl = 275 m2 Or, 227×72× l = 275 m2 Or, l=27511=25 Also, l2 = h2 + r2 Or, h=l2-r2h=252-722h=612.72=24.75 m Volume of tent made for two students =13πr2h=13×227×722×24.75=317.625 m3 (ii) Ratio of the volumes of tents =volume of the tent made for 2 studentsvolume of the tent made for 4 students To find the ratio of the volumes, we need to find the volume of tent made for four students. For that, we need to find the height of the tent. Area of the curved surface of a cone = πrl r is the radius of the cone with slant height l and height h. Here , πrl = 550 m2

Or, 227× 7 × l = 550 m2 Or, l=55022=25 m Also, l2 = h2 + r2 Or, h=l2-r2h=252-72h=576=24 m Volume of the tent made for four students =13πr2h=13×227×(7)2 ×24 m2 = 1232 m3 Volume of the tent made for two students = 317.625 m2 = 318 m3(approximately) Ratio of the volumes of tents=volume of the tent made for 2 studentsvolume of the tent made for 4 students=3181232=159616 (b) Total volume of the tents made for two students(V2) = 2 × 317.625 m2 = 635.25 m3 Volume of the tent made for four students (V4) = 1232 m3 The total volume of the two tents made for two students is less than the volume of the tent made for four students Difference between V4 and V2 = 1232 m3 - 635.25 m3 = 596.75 m3 Percentage of decrease in the volume = 596.751232×100=48.44 %