1. The cost (in dollars) of manufacturing one item is given by
C(x, y) = 30 + 3x + 5y
where x is the cost of 1 hour of labor and y is the cost of 1 pound of material.
(a) If the hourly cost of labor is $20, and the material costs $3 per pound, what is the cost of manufacturing one of these items?
(b) Find and interpret the partial derivative of C with respect to x.
2. The manufacture of 1 unit of a product has a cost (in dollars) given by
C(x, y, z) = 10 + 8x + 3y + z
where x is the cost of 1 pound of one raw material, y is the cost of 1 pound of a second raw material, and z is the cost of 1 work-hour of labor.
(a) If the cost of the first raw material is $16 per pound, the cost of the second raw material is $8 per pound, and labor costs $18 per work-hour, what will it cost to produce 1 unit of the product?
(b) Find and interpret the partial derivative of C with respect to x.
3. The total cost of producing 1 unit of a product is
C(x, y) =30 + 2x + 4y + xy/50 dollars
where x is the cost per pound of raw materials and y is the cost per hour of labor.
(a) If labor costs are held constant, at what rate will the total cost increase for each increase of $1 per pound in material cost?
(b) If material costs are held constant, at what rate will the total cost increase for each $1 per hour increase in labor costs?