The figure shows two sections of an old pipe system that runs through a hill, wi

Question

The figure shows two sections of an old pipe system that runs through a hill, with distances dA = dB = 30.0 m andD = 125 m. On each side of the hill, the pipe radius is 1.80 cm. However, the radius of the pipe inside the hill is no longer known.To determine it, hydraulic engineers first establish that water flows through the left and right sections at 2.90 m/s. Then they release a dye in the water at point A and find that it takes 97.0 s to reach point B. What is the average radius (in cm) of the pipe within the hill?

Explanation / Answer

distances dA=dB= 30.0 M
           D=125m

radius of pire =1.8cm
speed of flow v1=v2=2.90 m/s
total time taken t= 97.0 s

let average radius of the hill be R

the value of x = D-(dA+dB)
               =125-(30+30)
               =65 m

let t1,t2 and t3 be the time taken to travel distances dA,dB,x

t1=dA/v1
=30/2.9
=10.345 s

t2 = dB/v2
=30/2.9
=10.345 s

now

t=t1+t2+t3
t3=t-(t1+t2)
=97-(10.345+10.345)
=76.31

speed of water to travel distance x is

v3= x/t3

=65/76.31
=0.851 m/s

we have average radius R =sqrt((r^2*v1)/v3)
                         =sqrt((1.8^2*2.9)/0.851)
                         =3.323 cm

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