The volume of a cylinder is given by \(V=\pi r^{2} h\), where \(r\) is the radius of the cylinder and \(h\) is the height. Assume the radius, in \(\mathrm{cm}\), is lognormal with parameters \(\mu_{r}=1.6\) and \(\sigma_{r}^{2}=0.04\), the height, in \(\mathrm{cm}\), is lognormal with parameters \(\mu_{h}=1.9\) and \(\sigma_{h}^{2}=0.05\), and that the radius and height are independent.

a. Show that \(V\) is lognormally distributed, and compute the parameters \(\mu_{V}\) and \(\sigma_{V}^{2}\). (Hint: \(\ln V=\ln \pi+\) \(2 \ln r+\ln h\).)

b. Find \(P(V<600)\).

c. Find \(P(500

d. Find the mean of \(V\).

e. Find the median of \(V\).

f. Find the standard deviation of \(V\).

g. Find the 5th percentile of \(V\).

h. Find the 95th percentile of \(V\).