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:

(ii)} Find the derivative of the area \( A \) with respect to the height on the wall \( x \), and find its critical point.

Answer 1

To find the derivative of the area \( A \) with respect to \( x \), we apply the product and chain rule: \[ A(x) = \frac{1}{2} \times x \times \sqrt{h^2 - x^2} \] Differentiating with respect to \( x \): \[ \frac{dA}{dx} = \frac{1}{2} \times \left[ \sqrt{h^2 - x^2} + x \times \frac{d}{dx} \left( \sqrt{h^2 - x^2} \right) \right] \] Now, using the chain rule on the second term: \[ \frac{d}{dx} \left( \sqrt{h^2 - x^2} \right) = \frac{-2x}{2\sqrt{h^2 - x^2}} = \frac{-x}{\sqrt{h^2 - x^2}} \] Thus: \[ \frac{dA}{dx} = \frac{1}{2} \times \left[ \sqrt{h^2 - x^2} - \frac{x^2}{\sqrt{h^2 - x^2}} \right] \] Simplifying: \[ \frac{dA}{dx} = \frac{h^2 - 2x^2}{2\sqrt{h^2 - x^2}} \] To find the critical point, set \( \frac{dA}{dx} = 0 \): \[ h^2 - 2x^2 = 0 \] \[ x^2 = \frac{h^2}{2} \] \[ x = \frac{h}{\sqrt{2}} \] So, the critical point occurs when \( x = \frac{h}{\sqrt{2}} \).