In an annealing process, a 50-mm-thick stainless steel plate (rho = 8238 kg/m3, cp = 468 j/kg·k, k = 13.4 w/m·k, and α = 3.48 × 10−6 m2/s ) was reheated in a furnace from an initial uniform temperature of 230°c. the ambient temperature inside the furnace is uniform at 1000°c, and the convection heat transfer coefficient is 215 w/m2·k. if the entire stainless steel plate is to be heated to at least 600°c, determine the time that the plate should be heated in the furnace using the analytical one-term approximation method.

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .