Let's break down the problem step by step. ### **Given:** - Velocity profile for laminar flow: \[ u = U \left[ 1 - \left(\frac{r}{R}\right)^2 \right] \] Where: - \( u \) = local velocity at radius \( r \) - \( U \) = maximum velocity at the center - \( R \) = pipe radius - \( r \ = radial position - Pipe radius, \( R =\,\text{cm} = .2\,\text{m} \) - \( r = 15\,\text{cm} = .15\,\text{m} \) - Mean flow velocity, \( U_{mean} = 10\,\text{m/s} \) ### **Relationship between Mean and Maximum Velocity (laminar flow in a pipe):** \[ U_{mean} = \frac{U}{2} \implies U = 2 \times U_{mean} = 2 \times 10 = 20\,\text{m/s} \] ### **Calculate the local velocity at \( r = .15\,\text{m} \):** \[ u = U \left[ 1 - \left(\frac{r}{R}\right)^2 \right] \] \[ u = 20 \left[ 1 - \left(\frac{.15}{.2}\right)^2 \right] \] \[ \frac{.15}{.2} = .75 \] \[ (.75)^2 = .5625 \] \[ u = 20 \left[ 1 - .5625 \right] = 20 \times .4375 = 8.75\,\text{m/s} \] --- ### **Check the Options:** - 11.75 m/s - 12.95 m/s - 10.25 m/s - 16.85 m/s **None of the options matches the calculated value (8.75 m/s).** **Let's double-check the formula for mean velocity:** - For a parabolic velocity profile: \[ U_{mean} = \frac{1}{2} U_{max} \implies U_{max} = 2 \times U_{mean} = 20\,\text{m/s} \] The calculation is correct. The answer should be **8.75 m/s**, but it is not present among the options. #### **Conclusion** **Based on the provided options and calculation, none of the options are correct.** But if you must choose the closest, **10.25 m/s** is the nearest. --- ### **Final Answer:** > **Closest Option: 10.25 m/s** **But the correct calculated value is 8.75 m/s.**