Subscribe to our Youtube Channel - https://you.tube/teachoo

Last updated at Jan. 7, 2020 by Teachoo

Transcript

Example 43 A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan–1 (0.5). Water is poured into it at a constant rate of 5 cubic meter per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m. Water tank is in shape of cone Let r be the radius of cone, h be the height of cone, & α be the semi−vertical angle Given α=tan^(−1)(0.5) So, tan α =(0.5) 𝑟/ℎ = 0.5 𝑟/ℎ = 1/2 𝑟 = ℎ/2 Also, Water is poured at a constant rate of 5 cubic meter per hour So, 𝑑𝑉/𝑑𝑡=5 𝑚^3/ℎ𝑟 Where V is volume of cone Now, 𝑉=1/3 𝜋𝑟^2 ℎ 𝑉=1/3 𝜋(h/2)^2 ℎ (As 𝑟 = ℎ/2 from (1)) 𝑉=1/3 𝜋(〖h/4〗^2 )ℎ 𝑉=(𝜋ℎ^3)/12 Differentiate w.r.t.t 𝑑𝑉/𝑑𝑡=𝑑((𝜋ℎ^3)/12)/𝑑𝑡 𝑑𝑉/𝑑𝑡=𝜋/12 . (𝑑ℎ^3)/𝑑𝑡 𝑑𝑉/𝑑𝑡=𝜋/12 . (𝑑ℎ^3)/𝑑ℎ .𝑑ℎ/𝑑𝑡 𝑑𝑉/𝑑𝑡=𝜋(3ℎ^2 )/12 . 𝑑ℎ/𝑑𝑡 𝑑𝑉/𝑑𝑡=(𝜋ℎ^2)/4 . 𝑑ℎ/𝑑𝑡 𝑑ℎ/𝑑𝑡=20/(𝜋ℎ^2 ) We need to find, rate at which level of water is rising when depth is 4 m i.e. ├ 𝑑ℎ/𝑑𝑡┤|_(ℎ = 4𝑚) Putting h = 4m in (3) ├ 𝑑ℎ/𝑑𝑡┤|_(ℎ = 4𝑚)=20/(𝜋(ℎ)^2 )=20/16 × 1/𝜋=5/4 × 1/(22/7)=5/4 × 7/22=35/88 Hence, rate of change of water level is 𝟑𝟓/𝟖𝟖 m/hr.

Examples

Example 1

Example 2

Example 3

Example 4 Important

Example 5

Example 6

Example 7

Example 8 Important

Example 9 Important

Example 10

Example 11 Important

Example 12

Example 13 Important

Example 14

Example 15

Example 16

Example 17 Important

Example 18

Example 19

Example 20

Example 21

Example 22

Example 23

Example 24

Example 25

Example 26

Example 27

Example 28 Important

Example 29

Example 30 Important

Example 31

Example 32 Important

Example 33 Important

Example 34

Example 35 Important

Example 36

Example 37 Important

Example 38 Important

Example 39

Example 40 Important

Example 41 Important

Example 42 Important

Example 43 Important You are here

Example 44 Important

Example 45 Important

Example 46 Important

Example 47 Important

Example 48 Important

Example 49

Example 50 Important

Example 51

Chapter 6 Class 12 Application of Derivatives

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.