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

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  1. Chapter 6 Class 12 Application of Derivatives
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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 2cm/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 & length of ladder is 5 m i.e. AB = 5 & OB be the wall & OA be the ground. Let OA = ๐‘ฅ & OB = ๐‘ฆ Given that The bottom of the ladder is pulled along The ground, away from the wall at the rate of 2 cm/ s i.e. ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ก๏ทฏ = 2cm/sec We need to calculate at which rate height of ladder on the wall. Decreasing when foot of the ladder is 4 m away from the wall i.e. we need to calculate ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ when ๐‘ฅ = 4 cm. Wall OB is perpendicular to the ground OA using Pythagoras theorem (OB)2 + (OA)2 = (AB)2 ๐‘ฆ2 + ๐‘ฅ2 = (5)2 ๐‘ฆ2 + ๐‘ฅ2 = 25 Different w. r. t time ๐‘‘(๐‘ฆ2+๐‘ฅ2)๏ทฎ๐‘‘๐‘ก๏ทฏ = ๐‘‘(25)๏ทฎ๐‘‘๐‘ก๏ทฏ ๐‘‘ ๐‘ฆ2๏ทฏ๏ทฎ๐‘‘๐‘ก๏ทฏ + ๐‘‘(๐‘ฅ2)๏ทฎ๐‘‘๐‘Ÿ๐‘ก๏ทฏ = 0 ๐‘‘(๐‘ฆ2)๏ทฎ๐‘‘๐‘ก๏ทฏ ร— ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฆ๏ทฏ + ๐‘‘(๐‘ฅ2)๏ทฎ๐‘‘๐‘ก๏ทฏ ร— ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ฅ๏ทฏ = 0 2๐‘ฆ ร— ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ + 2๐‘ฅ ร— ๐‘‘๐‘ฅ๏ทฎ๐‘‘๐‘ก๏ทฏ = 0 2๐‘ฆ ร— ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ฅ๏ทฏ + 2๐‘ฅ ร— (2) = 0 2๐‘ฆ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ + 4๐‘ฅ = 0 2๐‘ฆ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โ€“ 4 ๐‘ฅ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โˆ’ 4๐‘ฅ๏ทฎ2๐‘ฆ๏ทฏ We need to find ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ when ๐‘ฅ = 4 cm ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘ฅ = 4๏ทฏ โˆ’ 4 ร— 4๏ทฎ2๐‘ฆ๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ๏ทฏ๏ทฎ๐‘ฅ = 4๏ทฏ โˆ’ 16๏ทฎ2๐‘ฆ๏ทฏ ๐‘ฅ2 + ๐‘ฆ2 = 25 Putting ๐‘ฅ = ๐‘ฆ (4)2 + ๐‘ฆ2 = 25 ๐‘ฆ2 = 25 โ€“ 16 ๐‘ฆ2 = 9 ๐‘ฆ = 3 Now we need to find ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ at y = 3 ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โˆ’ 16๏ทฎ2 ร— 3๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โˆ’ 8๏ทฎ3๏ทฏ Since length of OB is in cm & time is in sec So, ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โˆ’ 8 ๐‘๐‘š๏ทฎ3 ๐‘ ๐‘’๐‘๏ทฏ ๐‘‘๐‘ฆ๏ทฎ๐‘‘๐‘ก๏ทฏ = โˆ’ ๐Ÿ–๏ทฎ๐Ÿ‘๏ทฏ cm/sec Hence, the height of the ladder on the wall is decreasing at the rate of 8๏ทฎ3๏ทฏ cm/sec

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