Chapter 14.4, Problem 29E

### 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 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)=2x2âˆ’2xy+y2âˆ’7xâˆ’10y+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 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 cost and revenue in thousands of dollars are C(x,y)=2x2âˆ’2xy+y2âˆ’7xâˆ’10y+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)âˆ’C(x,y).

Substitute 5x+8y for R(x,y), and 2x2âˆ’2xy+y2âˆ’7xâˆ’10y+11 for C(x,y) in P(x,y)=R(x,y)âˆ’C(x,y).

P(x,y)=5x+8yâˆ’(2x2âˆ’2xy+y2âˆ’7xâˆ’10y+11)=12x+18yâˆ’2x2+2xyâˆ’y2âˆ’11

Thus, the profit function is P(x,y)=12x+18yâˆ’2x2+2xyâˆ’y2âˆ’11.

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

âˆ‚Pâˆ‚x=âˆ‚âˆ‚x(12x+18yâˆ’2x2+2xyâˆ’y2âˆ’11)=12âˆ’4x+2y

Equate the first derivative of function P(x,y)=12x+18yâˆ’2x2+2xyâˆ’y2âˆ’11 with respect to x to zero.

12âˆ’4x+2y=0âˆ’2x+y=âˆ’6y=âˆ’6+2x

The first partial derivative of function P(x,y)=12x+18yâˆ’2x2+2xyâˆ’y2âˆ’11 with respect to y by holding x constant

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started