Chapter 14.4, Problem 25E

### 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

# Profit Suppose that a manufacturer produces two brands of a product, brand 1 and brand 2. Suppose the demand for brand 1 is   x   = 70   − p 1 thousand units and the demand for brand 2 is y =   80   − p 2 thousand units, where p 1  and  p 2 are prices in dollars. If the joint cost function is C   =   x y in thousands of dollars, how many of each brand should be produced to maximize profit? What is the maximum profit?

To determine

To calculate: The number of units of each brand purchased to maximize the profit and the maximum profit. Suppose that a manufacturer produces two brands of a product, brand 1 and brand 2. Suppose the demand for brand 1 is x=70p1 thousand units and the demand for brand 2 is y=80p2 thousand units, where p1 and p2 are prices in dollars. The joint cost function is C=xy in thousands of dollars.

Explanation

Given Information:

The demand for brand 1 is x=70âˆ’p1 thousand units and the demand for brand 2 is y=80âˆ’p2 thousand units, where p1 and p2 are prices in dollars. The joint cost function is C=xy in thousands of dollars.

Formula used:

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

(1) Find the partial derivatives âˆ‚zâˆ‚x and âˆ‚zâˆ‚y.

(2) Find the critical points, that is, the point(s) that satisfy âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0.

(3) Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)âˆ’(zxy)2=âˆ‚2zâˆ‚x2â‹…âˆ‚2zâˆ‚y2âˆ’(âˆ‚2zâˆ‚xâˆ‚y)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=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

(2) When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

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.

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 demand for brand 1 is x=70âˆ’p1 for brand 2 is y=80âˆ’p2 in thousand units, the joint cost function is C=xy in thousands of dollars.

The profit function is given by P(x,y)=p1x+p2yâˆ’C(x,y).

The demand for brand 1 is x=70âˆ’p1 thousand units.

Rewrite x=70âˆ’p1 in terms of p1. So,

p1=70âˆ’x

The demand for brand 2 is y=80âˆ’p2 thousand units.

Rewrite y=80âˆ’p2 in terms of p2. So,

p2=80âˆ’y

Also, the joint cost function is C(x,y)=xy in thousands of dollars.

Substitute 70âˆ’x for p1, 80âˆ’y for p2, and xy for C(x,y) in P(x,y)=p1x+p2yâˆ’C(x,y).

P(x,y)=(70âˆ’x)x+(80âˆ’y)yâˆ’xy=70xâˆ’x2+80yâˆ’y2âˆ’xy

Thus, the profit function is P(x,y)=70xâˆ’x2+80yâˆ’y2âˆ’xy.

Use the power of x rule for derivatives, the constant function rule, and the coefficient rule,

Thus,

âˆ‚Pâˆ‚x=070âˆ’2xâˆ’y=0âˆ’y=2xâˆ’70y=70âˆ’2x

And,

âˆ‚Pâˆ‚y=080âˆ’2yâˆ’x=0x+2y=80

Now, calculate value of x and y

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