The velocity of a blood corpuscle in a vessel depends on how far the corpuscle is from the center of the vessel. Let R be the constant radius of the vessel; Vm, the constant maximum velocity of the corpuscle; r, the distance from the center to a particular blood corpuscle (variable); and Vr, the velocity of that corpuscle. The velocity Vr is related to the distance r by the equation Vr=Vm (1-r^2 /R^2). Find r when Vr=1/4Vm.
Vr = Vm * ( 1 - r ^ 2 / R ^ 2 )
Vr = ( 1 / 4 ) Vm
( 1 - r ^ 2 / R ^ 2 ) = 1 / 4 Multiply both sides by 4
4 * ( 1 - r ^ 2 / R ^ 2 )= 1
4 - 4 * r ^ 2 / R ^ 2 = 1 Subtract 4 to both sides
4 - 4 * r ^ 2 / R ^ 2 - 4 = 1 - 4
- 4 * r ^ 2 / R ^ 2 = - 3 Multiply both sides by - 1
4 * r ^ 2 / R ^ 2 = 3 Multiply both sides by R ^ 2
4 * r ^ 2 = 3 * R ^ 2 Divide both sides by 4
r ^ 2 = 3 R ^ 2 / 4
r = sqrt ( 3 R ^ 2 / 4 )
r = sqrt ( 3 ) * R / 2
To find the value of r when Vr = 1/4Vm, we can solve the equation Vr = Vm(1 - r^2/R^2) for r.
Given:
Vr = 1/4Vm
Vr = Vm(1 - r^2/R^2)
Now, substitute Vr with 1/4Vm:
1/4Vm = Vm(1 - r^2/R^2)
Next, simplify the equation:
1/4 = 1 - r^2/R^2
Now, isolate the r^2/R^2 term:
r^2/R^2 = 1 - 1/4
Simplify the right side:
r^2/R^2 = 3/4
To eliminate the fraction, multiply both sides by R^2:
r^2 = (3/4)R^2
Now, take the square root of both sides to isolate r:
r = √((3/4)R^2)
Simplify further:
r = (√3/2)R
Therefore, when Vr = 1/4Vm, the value of r is (√3/2)R.