Answer:
see below
Step-by-step explanation:
We know that we have a right triangle with x^2 + y^2 = 25^2
x^2 + y^2 =625
Taking the derivative with respect to time on each side
d/dt( x^2 + y^2) =d/dt 625
The derivative of x^2 with respect to t is 2 x times dx/dt and
the derivative of y^2 with respect to t is 2 y times dy/dt and
and the derivative of a constant is zero
2 x dx/dt + 2 y dy/dt = 0
We are trying to find dy/dt or the rate it is sliding down the wall
Subtracting 2y dy/dt
2 x dx/dt = - 2 y dy/dt
Divide each side by -2y
2x/- 2y * dx/dt = dy/dt
-x/y * dx/dt = dy/dt
We know that dx/dt = 2
If the base is 7 x^2 + y^2 =625 7^2 + y^2 = 625 so y = sqrt(625 - 49) =24
-7/24 * 2 = dy/dt
-7/12 ft/sec= dy/dt when x=7
If the base is 15 x^2 + y^2 =625 15^2 + y^2 = 625 so y = sqrt(625 - 225) =20
-15/20 * 2 = dy/dt
-3/2ft/sec= dy/dt when x=15
If the base is 24 x^2 + y^2 =625 24^2 + y^2 = 625 so y = sqrt(625 - 576) =7
-24/7 * 2 = dy/dt
-48/7 ft/sec= dy/dt when x=24
Now we need to find the rate at which the area is changing
A = 1/2 xy
Taking the derivative of each side
dA/dt =d/dt ( 1/2 xy)
Using the product rule of derivatives
= 1/2 ( x dy/dt + y dx/dt)
Using the information for 7 ft from the wall
x = 7, dy/dt = -7/12, y = 24 and dx/dt =2
= 1/2 ( 7 * -7/12 + 24*2)
= 1/2 ( -49/12+ 48)
= 527/24 ft^2/ sec
