A15 foot ladder is leaning against a wall of a house. the base of the ladder is pulled away from the wall at a rate of 3 ft/sec. find the rate at which the area of the triangle is changing when the base of the ladder is 9 ft from the wall.

7.875 ft/s

Explanation:

L = 15 ft

dx/dt = 3 ft/s

x = 9 ft

Let the top of ladder is coming down with the rate of dy/dt.

use Pythagorean theorem

L^2 = x^2 + y^2     (1)

Differentiate both sides with respect to t

0 = 2 x dx/dt + 2y dy/dt

x dx/dt = - y dy/dt

When, x = 9 ft then y = ? . Put this in equation (1)

15^2 = 9^2 + y^2

225 - 81 = y^2

y  = 12 ft

So

dy/dt = - x (dx/dt) / y = - 9 (3) / 12 = - 9/4 ft/s

Let A be the area of the triangle

A = 1/2 (base)(height)

A = 1/2 (x y)

Differentiate both sides with respect to t

dA/dt = 0.5 (y dx/dt + x dy/dt)

dA/dt = 0.5 [ 12 (3) - 9 (9/4)]

dA/dt = 0.5(36 - 81 /4) = 31.5 / 4 = 7.875 ft/s

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min