   Chapter 14.4, Problem 29E Mathematical Applications for the ...

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

Solutions

Chapter
Section Mathematical Applications for the ...

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

Profit A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are C ( x ,   y )   =   2 x 2 − 2 x y + y 2 −   7 x − 10 y + 11 R ( x , y )   =   5 x + 8 y Find the number of each type of product that should be manufactured to maximize profit. What is the maximum profit?

To determine

To calculate: The number of each type of product that should be manufactured to maximize the profit and the maximum profit. A company manufactures two products, A and B. If x is the number of thousands of units of A and y is the number of thousands of units of B, then the cost and revenue in thousands of dollars are C(x,y)=2x22xy+y27x10y+11 and R(x,y)=5x+8y.

Explanation

Given Information:

The cost and revenue in thousands of dollars are C(x,y)=2x22xy+y27x10y+11 and R(x,y)=5x+8y, if x is the number of thousands of units of A and y is the number of thousands of units of B.

Formula used:

To calculate relative maxima and minima of the function z=f(x,y),

(1) Find the partial derivatives zx and zy.

(2) Find the critical points, that is, the point(s) that satisfy zx=0 and zy=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)(zxy)2=2zx22zy2(2zxy)2.

(a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0.

(b) If D<0, then neither a relative maximum nor a relative minimum occurs.

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 and 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 x is denoted by fx and with respect to y is denoted by fy.

For a function z(x,y), the second partial derivative,

(1) When both derivatives are taken with respect to x is zxx=2zx2=x(zx).

(2) When both derivatives are taken with respect to y is zyy=2zy2=y(zy).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy).

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.

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 cost and revenue in thousands of dollars are C(x,y)=2x22xy+y27x10y+11 and R(x,y)=5x+8y, if x is the number of thousands of units of A and y is the number of thousands of units of B.

The profit function is given by P(x,y)=R(x,y)C(x,y).

Substitute 5x+8y for R(x,y), and 2x22xy+y27x10y+11 for C(x,y) in P(x,y)=R(x,y)C(x,y).

P(x,y)=5x+8y(2x22xy+y27x10y+11)=12x+18y2x2+2xyy211

Thus, the profit function is P(x,y)=12x+18y2x2+2xyy211.

Differentiate P=f(x,y) with respect to x by holding y constant,

Px=x(12x+18y2x2+2xyy211)=124x+2y

Equate the first derivative of function P(x,y)=12x+18y2x2+2xyy211 with respect to x to zero.

124x+2y=02x+y=6y=6+2x

The first partial derivative of function P(x,y)=12x+18y2x2+2xyy211 with respect to y by holding x constant

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