A railcar barge is a perfect wall$-$sided rectangle $52$m long, $9$m wide and floats initially at an even$-$keel draft of $2$m with a KG of $3$m in seawater with a density of $1025$ kg$/$m$^3.$ A $25,000$ kg railroad car is loaded onto the barge, with its center of mass $6$m in from the stern end of the barge. Use the small weights method for this problem, approximating the barges KG$,$ TPC$,$ MCT$1,$ and waterplane area as constant during the loading of the railcar.

a$.$ Calculate the TPC and the true $($GM based$)$ and approximate $($BM based$)$ moment to trim $1$cm for the barge and compare them.

Aw $=$ Waterplane area

b$.$ What is the parallel sinkage when the railcar comes aboard?

c$.$ What moment does the railcar make about the LCF of the vessel, and what is the change in trim using the exact moment to change trim $1$ cm$?$

d$.$ What are the new drafts at: the bow, the stern, amidship, and $10$m forward of amidships?