water is being siphoned from a cylindrical tank of radius 10m into a rectangular tank whose base measures 20m by 15m. if the depth of the water in the cylindrical tank in decreasing at a rate of 2m/s, at what rate is the depth of the water increasing in the rectangular tank
the ratio of the cross-sections is
300/100π = 3/π
so, the rectangular tank fills up π/3 times as fast, at a rate of 2π/3 m/s
To find the rate at which the depth of water is increasing in the rectangular tank, we can start by finding the relationship between the two tanks.
The volume of water in the cylindrical tank is given by the formula Vcyl = πr^2h, where r is the radius and h is the depth of the water.
The volume of water in the rectangular tank is given by the formula Vrect = lwh, where l is the length, w is the width, and h is the depth of the water.
Initially, when the height of water in the cylindrical tank is decreasing, we have the relation:
Vcyl = Vrect
Using the given dimensions, we can write:
π(10^2)h = (20)(15)h
Simplifying the equation gives:
100πh = 300h
Dividing both sides by h, we get:
100π = 300
Now, since the volume of water in the cylindrical tank remains constant, we can take the derivative of the equation with respect to t (time) to find the rate of change:
100π(dh/dt) = 0
Since the volume of the cylindrical tank is not changing, the rate of change of the height of the water in the cylindrical tank (dh/dt) is zero.
Now, let's find the rate at which the depth of water is increasing in the rectangular tank. We know that the volume of the rectangular tank is given by Vrect = (20)(15)h, and we want to find dh/dt.
Taking the derivative of Vrect with respect to t, we get:
(dVrect/dt) = (20)(15)(dh/dt)
We know that (dVrect/dt) = 0, as the volume of the rectangular tank remains constant. Therefore, we can solve for (dh/dt):
0 = (20)(15)(dh/dt)
Simplifying the equation gives:
0 = 300(dh/dt)
Dividing both sides by 300, we get:
0 = dh/dt
Therefore, the rate at which the depth of the water is increasing in the rectangular tank is 0 m/s. The depth of the water in the rectangular tank is not changing.