Calc rate problem?

Question

a ladder that is 25 ft is against a wall of a house. base of the ladder is sliding at a rate of 2 ft per second. If the ladder is 7 ft from the wall. At what rate is the area changing?

I got -.58 for vertical change, and the horizontal change is 2 ft. Show me how to do it.

Answer 1

well, ladder, ground(x) and the building side(y) make right angle triangle where area is

a=x*y/2

As x^2+y^2=25^2

y=sqrt(625-x^2)

rate of area changing with change of x would be:

da= (x*sqrt(625-x^2)' dx/2

when you find the derivative  you get

da=( (625-x^2- x)/sqrt(625-x^2) dx/2

with x being 7

da=((625-49-7)/sqrt(625-49)dx/2

da=569/24/2dx=11.852dx

so rate of the area change is 23.7ft^2