Chapter 14.3, Problem 5E

### 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 given by C ( x , y ) = 20 x + 70 y + x 2 1000 + x y 2 100 + dollars where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $10 per pound and the present hourly rate for labor is$24. How will an increase of(a) $1 per pound for raw materials affect the total cost?(b)$1 per hour in labor costs affect the total cost?

(a)

To determine

To calculate: The increase in total cost for each increase of $1 per pound of the raw material. The total cost of producing 1 unit of a product is given by C(x,y)=20x+70y+x21000+xy2100 dollars, where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is$10 per round and the present hourly rate for labor is $24. Interpret the answer. Explanation Given Information: The total cost of producing 1 unit of a product is given by C(x,y)=20x+70y+x21000+xy2100 dollars, where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is$10 per round and the present hourly rate for labor is $24. 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 f(x)=nxn1. Constant function rule for a constant c is such that, if f(x)=c then f(x)=0. The coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x). Calculation: Consider the problem, the total cost of producing 1 unit of a product is given by C(x,y)=20x+70y+x21000+xy2100 dollars, where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The provided function is C(x,y)=20x+70y+x21000+xy2100 (b) To determine To calculate: The increase in total cost for each increase of$1 per hour in labor costs. The total cost of producing 1 unit of a product is given by C(x,y)=20x+70y+x21000+xy2100 dollars, where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $10 per round and the present hourly rate for labor is$24. Interpret the answer.

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