The marginal rate of transformation (MRT) is indirectly related to marginal cost. The former deals primarily with economic priorities given available resources, while the latter is a purely quantitative figure dealing with the additional costs necessary to produce one more unit of something. These are two very different measures, but they are clearly linked.
MRT is a measure dealing primarily with opportunity costs. The simple concept is that to produce something given available resources, something else will have to be given up. MRT is the rate at which that “other” good, not produced, costs not to produce. In general, businesses strive to reach an equilibrium where the MRT is close to zero, but in periods of uncertainty, the MRT can add up to real cash and value. Therefore, the MRT deals with what the firm sacrifices when it decides to produce x instead of y.
Marginal cost (MC) is a simpler measure. It is merely the cost, given the preset amount of capital at your disposal, to make more of an item. If your present capital configuration is at half capacity, then your marginal costs to produce more items will be very low. If it is at full capacity, it will be very high, because both labor and machines will be strained to keep up with the demand. Therefore, MC deals with both available capital and the demand for the product. even if your current capital configuration is strained under the increased demand, that increased demand will drive up prices and can make up for any overtime.
MRT and MC are very different, but they have close linkages. For example, your firm makes both guitars and guitar strings. The marketing department has decided that, since the economy is depressed, focusing production on strings will make economic sense, because demand for strings might remain constant, while demand for the more expensive guitars will fall off. This kind of decision, while rational on the surface, must be made against a backdrop of available capital. Clearly, the same machines and workers who make guitar strings are not the same as those who make guitars. A shift here, especially a drastic one, will cost much, especially in the initial period of adjustment as old routines are disrupted.
MRT to MC
Decisions to shift production possibilities are always connected to both the availability of resources and the available capital. That is the clearest connection to MC. In the guitar firm example, the decision to shift production to strings because it is a much lower cost item will then -- all other things being equal -- add strain to the string-producing branch of the firm. MC will be much higher there until the requisite capital investments have been made to absorb the new production demands. There might be a condition where the shift might lead to MC increases that are so high that the shift is not economically worth it.