Suppose a vertical pipe is to be used as part of a system tomanually cycle nutrients upward from the floor of a lake. (Manylakes do this naturally, some do not. Green Lake, near Syracuse,NY, is one such lake.) A pump is to be installed on the lake floorat the base of the pipe. The base of the pipe will have a diameterof 9 cm. The nozzle of the pipe at the top will have a diameter of4 cm. The lake is 59 m deep at the installation point. (Ignore anyviscosity.)

f.)Write an equation that expresses the heat current H ofconduction through the ice as a function of the thickness h of theice sheet (of area A) already formed. Assume the air temperature isa constant -10 ?C, and that the temperature of the bottom of theice sheet is 0 ?C.

(g) (1 point) Since H = dQ dt , use your answer to (f) toexpress the amount of heat dQ conducted through the ice sheet intime dt.

(i) (2 points) Express the amount of heat dQ that must beremoved from the water at the bottom of the ice sheet to freeze themass dm you found in (g).

(j) (2 points) Based on (g) and (i), set the expressions for dQequal to each other to obtain a differential equation relating theheat that must be removed to freeze a new layer to the heatconducted through the ice sheet.

(k) (2 points) Separate variables, and integrate to find thethickness h of the ice sheet as a function of time t. (Note that h= 0 when t = 0