Question

A ladder of fixed length \( h \) is to be placed along the wall such that it is free to move along the height of the wall.
Based upon the above information, answer the following questions:

(i)} Express the distance \( y \) between the wall and foot of the ladder in terms of \( h \) and height \( x \) on the wall at a certain instant. Also, write an expression in terms of \( h \) and \( x \) for the area \( A \) of the right triangle, as seen from the side by an observer.

Answer 1

Let \( h \) represent the ladder's length, \( x \) the height it reaches on the wall, and \( y \) the distance from the wall to the ladder's base. By the Pythagorean theorem: \[ x^2 + y^2 = h^2 \quad \text{(1)} \] From equation (1), \( y \) can be expressed as: \[ y = \sqrt{h^2 - x^2} \] The area \( A \) of the right triangle formed by the ladder, wall, and ground is calculated as: \[ A = \frac{1}{2} \times x \times y = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \]