Question 7 - NCERT Exemplar - MCQs - Chapter 6 Class 12 Application of Derivatives (Term 1)

Last updated at Dec. 4, 2021 by Teachoo

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A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is:

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Question 7
A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is:
(A) 1/10 radian/sec (B) 1/20 radian/sec
(C) 20 radian/sec (D) 10 radian/sec
Let AB be the ladder
& OA be the wall & OB be the ground.
Given
Length of ladder is 5 m
AB = 5 m
AB = 500 cm
Let OA = π₯ cm, OB = π¦ cm & β π΄π΅π= π
Given that
Top of the ladder slides downwards at the rate of 10 cm/sec
i.e. π π/π π = β10 cm/sec
We need to calculate
at which the rate the angle between the floor and the ladder decreases when lower end of ladder is 2 metres from the wall.
i.e. We need to calculate π π" " /π π when π = 2 m or π=πππ ππ.
Now,
We need to find π π½/π π , so for that we will differentiate π₯ w.r.t. π‘, since value of π π/π π is given.
sinβ‘γ π=(πΆππππππππ πΊππ π)/π―πππππππππ " " γ
sinβ‘γ π=ππ΄/π΄π΅ " " γ
sinβ‘γ π=π₯/500 " " γ
π=πππ π¬π’π§β‘π½
cosβ‘γ π=(π¨π ππππππ πΊππ π)/π―πππππππππ " " γ
cosβ‘γ π=ππ΅/π΄π΅ " " γ
cosβ‘γ π=π¦/500 " " γ
π=πππ πππβ‘π½
Differentiating π w.r.t. π
π₯=500 sinβ‘π
π π/π π = 500 Γ (π(sinβ‘π))/ππ‘
ππ₯/ππ‘ = 500 Γ (π (πππβ‘π½))/π π½ Γ ππ/ππ‘
π π/π π = 500 Γ πππ β‘π Γ ππ/ππ‘
Putting π π/π π = -10 cm/sec
β10 = 500 cosβ‘π Γ ππ/ππ‘
(β10)/(500 πππ β‘π )=ππ/ππ‘
ππ/ππ‘= (β10)/(500 cosβ‘π )
ππ/ππ‘= (β1)/(50 cosβ‘π )
Putting ππ¨π¬β‘π½=π/πππ from equation (2)
π π½/π π= (β1)/(50 Γ π/πππ)
π π½/π π = (β1)/( π¦/10)
π π½/π π= (βππ)/( π)
Putting π=πππ
β π π½/π πβ€|_(π =πππ) " "=(β10)/200
β ππ/ππ‘β€|_(π¦ =200) " "=(βπ)/ππ
Hence, angle between ladder and the ground is decreasing at rate
of π/ππ radian/sec.
So, the correct answer is (B)

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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.