Chapter 14.3, Problem 6E

### 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 ) =   30   +   0.5 x 2 +   30 y   −   x y dollars where x is the hourly labor rate and y is the cost per pound of raw materials. The current hourly rate is $25, and the raw materials cost$6 per pound. How will an increase of(a) $1 per pound for the raw materials affect the total cost?(b)$1 in the hourly labor rate affect the total cost?

(a)

To determine

To calculate: The increase in total cost for each increase of $1 per pound of the raw materials. The total cost of producing 1 unit of a product is given by C(x,y)=30+0.5x2+30yxy dollars, where x is the hourly labor rate and y is the cost per pound of raw materials. The currently hour rate is$25, and the raw materials cost $6 per pound. Explanation Given Information: The total cost of producing 1 unit of a product is given by C(x,y)=30+0.5x2+30yâˆ’xy dollars, where x is the hourly labor rate and y is the cost per pound of raw materials. The currently hour rate is$25, and the raw materials cost $6 per pound. Formula used: The rate of change of a function f(x,y) with respect to a variable y is given by the partial derivative of f with respect to y. For a function f(x,y), the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by fy. 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). Calculation: Consider the problem, the total cost of producing 1 unit of a product is given by C(x,y)=30+0 (b) To determine To calculate: The increase in total cost for each increase of$1 hourly labor rate. The total cost of producing 1 unit of a product is given by C(x,y)=30+0.5x2+30yxy dollars, where x is the hourly labor rate and y is the cost per pound of raw materials. The currently hour rate is $25, and the raw materials cost$6 per pound. Interpret the answer.

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