Murray's Law for Plants This problem is based on McCulloh et al. (2003). The plant xylem is a transport network within plants that forms a network like the blood vessels of animals. The xylem transports water from the roots, up the plant stem to its leaves. Unlike blood vessels, in some plants the xylem vessels are not single tubes, but are made up of bundles of smaller tubes. Larger xylem vessels contain more tubes, smaller vessels contain fewer tubes. Because vessels are made of smaller tubes, the way that transport costs depend on vessel radius is different for the xylem than for the blood vessels of an animal. Specifically, it can be shown that the cost of transporting water at a flow rate f (mea- sured in milliliters/s) in a xylem vessel of radius r and length (both measured in cm) is given by the function = 0.071 se T(r) = 0.071 rre where ry is the radius of one of the tubes within the xylem vessel (you may assume that rt = 5 x 10-2 cm). and (a) Assume that the cost of building the xylem vessel is still pro- portional to its volume: M(r) = bare where b is the metabolic cost of building and maintaining 1 cm of xylem vessel. If the plant controls xylem vessel radius to min- imize the total cost T(m)+ Mr), derive a formula relating xylem radius r to flow rate f. Your formula will include b as an unknown coefficient. (b) If a xylem vessel of radius R branches into two smaller ves- sels of radii n and r2, and all vessels minimize the total cost of transport and maintenance, show that the xylem vessel radii are related by Murray's law for plants: R = p + 1 .