Two pipes with diameters $d$ and $2 d$ with water flowing through them at speeds $v$ and $v / 2$, respectively flow into a pipe with diameter $4 d$. What is the flow speed in the $4 d$ diameter pipe? $v / 4$ $3 v / 16$ $v / 8$ $5 v / 32$

Step 1 ：Given two pipes with diameters \(d\) and \(2d\) with water flowing through them at speeds \(v\) and \(v / 2\), respectively, flow into a pipe with diameter \(4d\). We are asked to find the flow speed in the \(4d\) diameter pipe.

Step 2 ：The flow rate of a pipe is given by the equation \(Q = Av\), where \(A\) is the cross-sectional area of the pipe and \(v\) is the velocity of the fluid.

Step 3 ：Since the flow rate is conserved, the sum of the flow rates in the two smaller pipes must equal the flow rate in the larger pipe. We can set up an equation to represent this and solve for the velocity in the larger pipe.

Step 4 ：Let's calculate the cross-sectional areas of the pipes. For the pipe with diameter \(d\), the area \(A1 = \frac{\pi d^2}{4}\). For the pipe with diameter \(2d\), the area \(A2 = \pi d^2\). For the pipe with diameter \(4d\), the area \(A3 = 4\pi d^2\).

Step 5 ：Next, we calculate the flow rates for each pipe. For the pipe with diameter \(d\), the flow rate \(Q1 = \frac{\pi d^2 v}{4}\). For the pipe with diameter \(2d\), the flow rate \(Q2 = \frac{\pi d^2 v}{2}\).

Step 6 ：Since the flow rate is conserved, we can set up the equation \(Q1 + Q2 = Q3\), where \(Q3\) is the flow rate in the larger pipe. Substituting the values we calculated, we get \(\frac{\pi d^2 v}{4} + \frac{\pi d^2 v}{2} = 3\frac{\pi d^2 v}{4}\).

Step 7 ：Solving for \(v3\), the velocity in the larger pipe, we get \(v3 = \frac{3v}{16}\).

Step 8 ：Final Answer: The flow speed in the \(4d\) diameter pipe is \(\boxed{\frac{3v}{16}}\).