Chapter 14.3, Problem 3E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# The total cost of producing 1 unit of a product is C ( x ,   y ) =   30   + 2 x   +   4 y   + x y 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, find the function that describes the rate at which total cost increases for each increase of $1 per pound in material cost.(b) If material costs are held constant, find the function that describes the rate at which total cost increases for each$1 per hour increase in labor costs.

(a)

To determine

To calculate: The function that describes the rate at which total cost increases for each increase of $1 per pound in material cost if labor costs are held constant. The total cost of producing 1 unit of a product is C(x,y)=30+2x+4y+xy50 dollars, where x is the cost per pound of raw materials and y is the cost per hour of labor. Explanation Given Information: The total cost of producing 1 unit of a product is C(x,y)=30+2x+4y+xy50 dollars, where x is the cost per pound of raw materials and y is the cost per hour of labor. Formula used: The rate of change of a function f(x,y) with respect to a variable x is given by the partial derivative of f with respect to x. For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant. The partial derivative of f with respect to x is denoted by fx. Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1. Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0. The coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x) (b) To determine To calculate: The function that describes the rate at which total cost increases for each$1 per hour increase in labor costs if material costs are held constant. The total cost of producing 1 unit of a product is C(x,y)=30+2x+4y+xy50 dollars, where x is the cost per pound of raw materials and y is the cost per hour of labor.

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