Q)  Adventure camps are the perfect place for the children to practice decision making for themselves without parents and teachers guiding their every move. Some students of a school reached for adventure at Sakleshpur. At the camp, the waiters served some students with a welcome drink in a cylindrical glass and some students in a hemispherical cup whose dimensions are shown below.

After that they went for a jungle trek. The jungle trek was enjoyable but tiring. As dusk fell, it was time to take shelter. Each group of four students was given a canvas of area 551 m² . Each group had to make a conical tent to accommodate all the four students. Assuming that all the stitching and wasting incurred while cutting, would amount to 1 m² , the students put the tents. The radius of the tent is 7m.

1. The volume of cylindrical cup is:

a) 295.75 cm³      b) 7415.5 cm³          c) 384.88 cm³           d) 404.25 cm³

2. The volume of hemispherical cup is:

a) 179.67 cm³       b) 89.83 cm³         c) 172.25 cm³        d) 210.60 cm³

3. Which container had more juice and by how much?

a) Hemispherical cup, 195 cm³                b) Cylindrical glass, 207 cm³

c) Hemispherical cup, 280.85 cm³         d) Cylindrical glass, 314.42 cm³

4. The height of the conical tent prepared to accommodate four students is

a) 18m     b) 10m      c) 24m    d) 14m

5. How much space on the ground is occupied by each student in the conical tent

a) 54 m²   b) 38.5 m²      c) 86 m²   d) 24 m²

Ans:

1. Cylindrical cup volume:

The volume of the cylindrical cup is = π r ² h

Given: D_cyl = 7 cm, ∴ R_cyl = cm; and H_cyl = 10.5 cm

∴ The volume of the cylindrical cup =

= x (10.5)

= 11 x x 10.5

= 404.25 cm³

Hence, option d) is the correct option.

2. Hemispherical cup volume:

The volume of hemispherical cup is =

Given: D_hemi = 7 cm, ∴ R_hem = cm

∴ The volume of the hemispherical cup =

=

=

=

=

= 89.83 cm³

Hence, option b) is the correct option.

3. Larger container:

As we can see that the volume of cylindrical cup is more than the hemispherical cup, hence cylindrical cup will hold more juice.

Since the difference in the capacity = capacity of the cylindrical cup – capacity of the hemispherical  cup

= 404.25 – 89.83

= 314.42 cm³

Therefore, the Cylindrical cup will hold 314.42 cm³ more juice.

Hence, option d) is the correct option.

4. Height of the tent:

Given: the area of the canvas = 551 m²

Wastage of canvas = 1 m²

∴ The net canvas used for making the conical tent = original area of canvas – wastage

= 551 – 1 = 550 m²

Now this net canvas is used to make a tent,

Next, the Curved Surface Area of a conical tent is = π r l

Given: the radius of the conical tent, R_con = 7 m

∴ 550 = π x 7 x L_con

∴ 550 = x 7 x L_con

∴ 550 = 22 x L_con

∴ L_con = = 25 m

Now, from the adjoining figure:

(H_con) ² + (R_con) ²  = (L_con) ²

∴ (H_con)² = (L_con)²  – (R_con)²

 Now, we already have, R_con = 7 m, L_con = 25 m

∴ (H_con)² = (25) ² – (7) ² = 625 – 49 = 576

∴ H_con = √ 576 = 24 m

Therefore, the height of the conical tent is 24 m

Hence, option c) is the correct option.

5. Ground space by each student in a conical tent:

Ground space of the conical tent = area of the base of the tent

∵ The base area of a conical tent = π r ²

and R_con = 7 m

The base area of a conical tent = π r ² = x (7)²

= 22 x 7 = 154 cm²

Since the tent can contain 4 students, hence the ground space will be equally shared by the 4 students.

Therefore, space covered by each student =  = 38.5 cm²

Hence, option b) is the correct option. 

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