For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor . The Fanning friction factor is dependent on a

For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor ƒ. The Fanning friction factor is dependent on a number of parameter related to the size of the pipe and the fluid, which can all be represented by another dimensionless quantity, the Reynolds number Re. A formula that predicts ƒ given Re is the von Karman equation,

1/√f = 4 log ₁₀ (Re √f) – 0.4

Typical values for the Reynolds number for turbulent flow are 10,000 to 500,000 and for the Fanning friction factor are 0.001 to 0.01. Develop a function that uses bisection to solve for ƒ given a user-supplied value of Re between 2,500 and 1,000,000. Design the function so that it ensures that the absolute error in the result is E_α.d < 0.000005.