Case Study based-2: In the month of December 2020, it rained heavily throughout the day over the city of Hyderabad. Anil observed the raindrops as they reached him. Each raindrop was in the shape of a hemisphere surmounted by a cone of the same radius of 1 mm. Volume of one of such drops is 3.14 mm³. Anil collected the rain water in a pot having a capacity of 1099 cm³. [Use √2 = 1.4]
Based on the above situation, answers the following questions. (a) Find the total height of the drop. (i) 1 mm (ii) 2 mm (iii) 3 mm (iv) 4 mm (b) The curved surface area of the drop is (i) 8.74 mm³ (ii) 9.12 mm³ (iii) 10.68 mm³ (iv) 12.54 mm³ (c) As the drop fell into the pot, it changed into a sphere. What was the radius of this sphere? (i) (3/4)1/3(ii) (4/3)1/3(iii) 31/3(iv) 41/3(d) How many drops will fill the pot completely. (i) 260000 (ii) 280000 (iii) 320000 (iv) 350000 (e) The total surface area of a hemisphere of radius r is (i) 2/3 ߨr3(ii) 4/3 ߨr3(iii) 2ߨr2(iv) 3ߨr2
Answers
Given : Each raindrop was in the shape of a hemisphere surmounted by a cone of the same radius of 1 mm
Volume of one of such drops is 3.14 mm³.
Anil collected the rain water in a pot having a capacity of 1099 cm³.
[ √2 = 1.4]
To Find :
(a) total height of the drop. (i) 1 mm (ii) 2 mm (iii) 3 mm (iv) 4 mm
(b) The curved surface area of the drop is (i) 8.74 mm² (ii) 9.12 mm² (iii) 10.68 mm²(iv) 12.54 mm²
(c) As the drop fell into the pot, it changed into a sphere. What was the radius of this sphere (i) ∛3/4 (ii) ∛(4/3) (iii) ∛3 (iv) ∛4
(d) How many drops will fill the pot completely. (i) 260000 (ii) 280000 (iii) 320000 (iv) 350000
(e) The total surface area of a hemisphere of radius r is
Solution:
Volume of Drop = Volume of cone + Volume of hemisphere
= (1/3)πr²h + (2/3)πr³
h = height of cone
r = 1 mm
Height of drop = h + r = h + 1 mm
(1/3)π(1)²h + (2/3)π(1)³ = 3.14
π = 3.14
=> h/3 + 2/3 = 1
=> h/3 = 1/3
=> h = 1
Height of drop = 1 + 1 = 2 mm
r = 1 , h = 1
slant height of cone = √r² + h² = √ 1² + 1² = √2 = 1.4
curved surface area of the drop = CSA of cone + CSA of hemisphere
= πrl + 2πr²
= π(1) 1.4 + 2π(1)²
= 3.4π
= 3.4 (3.14)
= 10.676
= 10.68 mm²
Volume of one of such drops is 3.14 mm³
Volume of sphere = (4/3)πr³ = 3.14
=> r³ = 3/4
=> r = ∛(3/4)
radius of this sphere = ∛(3/4)
1 cm = 10mm
1 cm³ = 1000 mm³
capacity of pot = 1099 cm³ = 1099 * 1000 mm³
Number of drops will fill the pot completely = 1099 * 1000 / 3.14
= 350000
The total surface area of a hemisphere of radius r is = 3πr²
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Answer:
Ans:
(a) (ii) 2 mm
(b) (iii) 10.68 mm³
(c) (i) (3/4)1/3
(d) (iv) 350000
(e) (iv) 3r
Step-by-step explanation: