Q) A ladder set against a wall at an angle 45° to the ground. If the foot of the ladder is pulled away from the wall through a distance of 4 m, its top slides a distance of 3 m down the wall making an angle 30° with the ground. Find the final height of the top of the ladder from the ground and length of the ladder.

Ans:  

VIDEO SOLUTION

STEP BY STEP SOLUTION

Let’s start by making a  diagram for the question:

Here, BC is the ladder, and when its top slides down 3 m along the wall, its bottom slides away by 4 m.

Let’s consider the final height of the top is h from the ground and the length of the ladder is L.

(i) Height of ladder’s top (h):

(before we start, we should keep it in mind that we have to find height h and hence, we will convert all equations in h form…

Let’s start from Δ ABC, Sin 45⁰ = 

∴

∴ L = (h + 3) √2 ….. (i)

In Δ ADE, Sin 30⁰ = 

∴

∴ L = 2 h………..….. (ii)

By comparing equations (i) and (ii), we get:

∴ (h + 3) √2 = 2 h

∴ 3 √2 = 2 h – h √2

∴ 3 √ 2 = h √2 (√2 – 1)

∴ 3  = h (√2 – 1)

∴ h =

∴ h =

∴ h =            [∵ (a + b) (a – b) = a² – b² ]

∴ h =

∴ h = 3 (√2 + 1) = 3 (1.41 +1) = 7.23 m

Therefore, the final height of the ladder’s top = 7.23 m

(ii) Length of the ladder (BC or DE):

The length of the ladder is given by BC or DE

From equation (ii), L = 2 h

∴ L = 2 x 7.23 = 14.46 m

OR

By transferring the value of h in the above equation, we get:

∴ L = 2 h = 2 x 3 (√2 + 1)

∴ L = 6 (√2 + 1) = 6 (1.41 + 1) = 14.46 m

Therefore, the length of the ladder is 14.46 m

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