To find the rate at which water is flowing through the pipe, we need to calculate the volume of water passing through each infinitesimally thin cylindrical section of the pipe and then integrate those volumes for the entire length of the pipe.
Let's consider an infinitesimally thin cylindrical section of the pipe with radius r and thickness dr. The volume of water passing through this section in one second can be calculated by multiplying the cross-sectional area of the section by the speed of the water at that radius.
The cross-sectional area of the cylindrical section is given by A = πr^2 (since the pipe has a constant radius of 1 inch).
The speed of the water at distance r from the center is given by v = 10(1 - r^2) inches per second.
The volume of water passing through the infinitesimally thin cylindrical section in one second is therefore given by dV = A * v * dt = πr^2 * (10(1 - r^2)) * dt.
Integrating this expression for V, the total volume of water passing through the entire length of the pipe, we get:
V = ∫[0,1] πr^2 * (10(1 - r^2)) * dr.
To calculate this integral, expand the expression and integrate term by term:
V = ∫[0,1] (10πr^2 - 10πr^4) * dr.
Integrating term by term:
V = [10Ï€r^3/3 - 10Ï€r^5/5] evaluated from 0 to 1.
Plugging in the upper and lower limits:
V = (10Ï€/3 - 10Ï€/5) - (0) = (10Ï€/3 - 2Ï€) cubic inches per second.
Therefore, the rate at which water is flowing through the pipe is (10Ï€/3 - 2Ï€) cubic inches per second.