A ladder inclined at 60o with the horizontal is leaning against a vertical wall.

Question

A ladder inclined at 60o with the horizontal is leaning against a vertical wall. The foot of the ladder is 3 𝑚. away from the foot of the wall. A boy climbs the ladder such that his distance 𝑧 𝑚𝑒𝑡𝑒𝑟𝑠 with respect to the foot of the ladder is given by 𝑧 = 6𝑡 , where 𝑡 is the time in seconds.

(𝑎) Find the rate at which his vertical distance from the ground changes with respect to 𝑡 .

Hint: Pythagorean Theorem

(𝑏) Find the rate at which his distance from the foot of the wall is changing with respect to 𝑡 when he is 3 𝑚. away from the foot of the ladder.

Hint: Cosine Law

Answer 1

a) Let y: vertical distance from the ground

By special right triangle: y = √3/2·z

dy/dt = √3/2·dz/dt ; with dz/dt = 6

dy/dt = 3√3 m/sec

b) Let f: distance to foot of wall

By Law of Cosines: f² = z² + 3² - 6zcos60 ; f² = z² - 3z + 9

2f·df/dt = 2z·dz/dt - 3·dz/dt ; when z = 3 then f = 3 also.

6df/dt = 18

df/dt = 3 m/sec