A chemical manufacturing plant can produce z units of chemical Z given p units of chemical P and r units of chemical R, where: z = 180p0.75 0.25 Chemical P costs $100 a unit and chemical R costs $500 a unit. The company wants to produce as many units of chemical Z as possible with a total budget of $80,000. A) How many units each chemical (P and R) should be "purchased" to maximize production of chemical Z subject to the budgetary constraint? Units of chemical P, p = Units of chemical R, r = B) What is the maximum number of units of chemical Z under the given budgetary conditions? (Round your answer to the nearest whole unit.) Max production, z= unitsFind the dimensions of a box with open top made out of 75 ft2 of cardboard so that it has the largest volume. (a) Assume that the dimensions of the bottom of the box are 2': ft and y ft, and the height of the box is 2: ft. If ay, z) is maximized, and 9(33, y, z) = 0 is the constraint, find functions f and g. flm,y,z)=[ } g($:ysz) =' ' (b) Find the gradients Vf and V9. Vas4=