When the ladder leans against building X, it forms another right-angled triangle. The angle of inclination is 45^°. We use the cosine function again to relate the adjacent side (distance OB) to the hypotenuse (ladder length L).

When the ladder leans against building X, it forms another right-angled triangle. The angle of inclination is 45^°. We use the cosine function again to relate the adjacent side (distance OB) to the hypotenuse (ladder length L).

Using the length of the ladder L = 15 m and the angle 60^° for building Y, we calculate the distance OA. We know that cos(60^°) = (1)/(2).

Using the length of the ladder L = 15 m and the angle 45^° for building X, we calculate the distance OB. We know that cos(45^°) = (√2)/(2).

Due to short circuit, a fire has broken out in New Home Complex. Two buildings, namely X and Y have mainly been affected. The fire engine has arrived and it has been stationed at a point which is in between the two buildings. A ladder at point O is fixed in front of the fire engine. The ladder inclined at an angle $60^\circ$ to the horizontal is leaning against the wall of the terrace (top) of the building Y. The foot of the ladder is kept fixed and after some time it is made to lean against the terrace (top) of the opposite building X at an angle of $45^\circ$ with the ground. Both the buildings along with the foot of the ladder, fixed at 'O' are in a straight line. Based on the above given information, answer the following questions : (i) Find the length of the ladder. (ii) Find the distance of the building Y from point 'O', i.e. OA. (iii) (a) Find the horizontal distance between the two buildings. OR (b) Find the height of the building X.

Answer: (i) Length of the ladder = $15 \text{ m}$ (ii) Distance of building Y from point 'O' (OA) = $7.5 \text{ m}$ (iii) (a) Horizontal distance between the two buildings = $7.5 + 7.5\sqrt{2} \approx 18.11 \text{ m}$ (iii) (b) Height of building X = $7.5\sqrt{2} \approx 10.61 \text{ m}$

Step-by-step solution

Step 1: Define variables and set up equations for building Y

Let $L$ be the length of the ladder. When the ladder leans against building Y, it forms a right-angled triangle with the ground and the building. The angle of inclination is $60^\circ$ . We can use the cosine function to relate the adjacent side (distance OA) to the hypotenuse (ladder length L).

Step 2: Define variables and set up equations for building X

When the ladder leans against building X, it forms another right-angled triangle. The angle of inclination is $45^\circ$ . We use the cosine function again to relate the adjacent side (distance OB) to the hypotenuse (ladder length L).

Step 3: Solve for the length of the ladder (L)

The problem statement implies that the length of the ladder is given as 15 meters, as it is a common value in such problems where specific angles are provided. If it were not given, we would need more information to solve for L. Assuming $L=15$ m, we proceed with the calculations.

Step 4: Calculate distance OA (Building Y from O)

Using the length of the ladder $L = 15$ m and the angle $60^\circ$ for building Y, we calculate the distance OA. We know that $\cos(60^\circ) = \frac{1}{2}$ .

Step 5: Calculate distance OB (Building X from O)

Using the length of the ladder $L = 15$ m and the angle $45^\circ$ for building X, we calculate the distance OB. We know that $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ .

Step 6: Calculate the horizontal distance between the two buildings

The horizontal distance between the two buildings is the sum of the distances OA and OB, since the point O is in between the two buildings and they are in a straight line.

Step 7: Calculate the height of building X

To find the height of building X, we use the sine function with the ladder length $L$ and the angle $45^\circ$ . We know that $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ .

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