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Finding rate of change

Ex 6.1, 10 - A ladder 5 m long is leaning against a wall

Ex 6.1,10 - Chapter 6 Class 12 Application of Derivatives - Part 2
Ex 6.1,10 - Chapter 6 Class 12 Application of Derivatives - Part 3
Ex 6.1,10 - Chapter 6 Class 12 Application of Derivatives - Part 4
Ex 6.1,10 - Chapter 6 Class 12 Application of Derivatives - Part 5


Transcript

Ex 6.1, 10 A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?Let AB be the ladder & OB be the wall & OA be the ground. Given Length of ladder is 5 m AB = 5 cm Let OA = π‘₯ cm & OB = 𝑦 cm Given that Bottom of ladder is pulled away from the wall at the rate of 2 cm/ s i.e. π’…π’š/𝒅𝒕 = 2 cm/sec We need to calculate at which rate height of ladder on the wall is decreasing when foot of the ladder is 4 m away from the wall i.e. We need to calculate 𝒅𝒙/𝒅𝒕 when π’š = 4 cm. Since Wall OB is perpendicular to the ground OA Using Pythagoras theorem (OB)2 + (OA)2 = (AB)2 𝑦2 + π‘₯2 = (5)2 π’šπŸ + π’™πŸ = 25 Differentiating w.r.t time (𝑑(𝑦2 + π‘₯2))/𝑑𝑑 = (𝑑(25))/𝑑𝑑 (𝑑 (𝑦2))/𝑑𝑑 + (𝑑(π‘₯2))/𝑑𝑑 = 0 (𝑑(𝑦2))/𝑑𝑑 Γ— 𝑑𝑦/𝑑𝑦 + (𝑑(π‘₯2))/𝑑𝑑 Γ— 𝑑π‘₯/𝑑π‘₯ = 0 (𝑑(𝑦2))/𝑑𝑦 Γ— 𝑑𝑦/𝑑𝑑 + (𝑑(π‘₯2))/𝑑π‘₯ Γ— 𝑑π‘₯/𝑑𝑑 = 0 2𝑦 Γ— π’…π’š/𝒅𝒕 + 2π‘₯ Γ— 𝑑π‘₯/𝑑𝑑 = 0 2𝑦 Γ— 2 + 2π‘₯ Γ— 𝑑π‘₯/𝑑𝑑 = 0 ("From (1): " 𝑑𝑦/𝑑𝑑= 2 m/s) 2π‘₯ 𝑑π‘₯/𝑑𝑑 = –4𝑦 𝒅𝒙/𝒅𝒕 = (βˆ’πŸπ’š)/𝒙 Putting 𝑦 = 4 cm β”œ 𝑑π‘₯/𝑑𝑑─|_(𝑦 = 4) =(βˆ’2 Γ— 4)/π‘₯ β”œ 𝒅𝒙/𝒅𝒕─|_(π’š = πŸ’) =(βˆ’πŸ–)/𝒙 To find 𝑑π‘₯/𝑑𝑑 , we need to find value of x for y = 4 Finding value of x We know that π‘₯2 + 𝑦2 = 25 Putting 𝑦 =4 π‘₯2 + 42 = 25 π‘₯2 = 25 – 16 π‘₯2 = 9 π‘₯ = 3 Putting x = 3 in (2) 𝑑π‘₯/𝑑𝑑=(βˆ’8)/π‘₯ 𝑑π‘₯/𝑑𝑑 = (βˆ’8)/3 Since x is in cm & time is in sec ∴ 𝑑π‘₯/𝑑𝑑 = (βˆ’πŸ–)/πŸ‘ cm/sec Hence, height of ladder on the wall is decreasing at rate of πŸ–/πŸ‘ cm/sec

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.