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 the length of the ladder be \( h \), the height of the ladder on the wall be \( x \), and the distance of the foot of the ladder from the wall be \( y \). Using the Pythagorean theorem, we have: \[ x^2 + y^2 = h^2 \quad \text{(1)} \] From equation (1), express \( y \) in terms of \( x \) and \( h \): \[ y = \sqrt{h^2 - x^2} \] Next, the area \( A \) of the right triangle formed by the ladder, wall, and ground is given by: \[ A = \frac{1}{2} \times x \times y = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \]