Question 2 - NCERT Exemplar - MCQs - Chapter 6 Class 12 Application of Derivatives

Last updated at April 16, 2024 by Teachoo

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:

Question 2
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)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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