Chapter 10.4, Problem 31E

### 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

# Minimum cost A printer has a contract to print 100,000 posters for a political candidate. He can run the posters by using any number of plates from 1 to 30 on his press. If he uses x metal plates, they will produce x copies of the poster with each impression of the press. The metal plates cost $20 to prepare, and it costs$125 per hour to run the press. If the press can make 1000 impressions per hour, how many metal plates should the printer make to minimize costs?

To determine

To calculate: The number of metal plates required to minimize the costs.

Explanation

Given Information:

x Metal plates chosen from 1-30 can produce x copies of the poster. The total number of posters required are 100,000. The preparation cost is $20 and$125 to run the press for an hour. The press can make 1000 impressions in an hour.

Formula used:

To find the minimum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is more than zero, then substituting the value of the independent variable will give the minimum value of the equation.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function f(x)=xn, the derivative is

ddx[xn]=nxn1

Calculation:

As given, the total number of posters required are 100,000.

x Metal plates chosen from 1-30 can produce x copies of the poster.

The preparation cost is $20 and$125 to run the press for an hour. The press can make 1000 impressions in an hour.

Let x is the number of plates and y is the number of impressions.

The cost is C=20x+125y1000

Use the equation of volume xy=100,000 and calculate y in terms of x,

y=100,000x

Substitute y=100,000x in C=20x+125y1000,

C=20x+1251000×100,000xC=20x+12500x

To find the stationary value for C=20x+125y1000, differentiate the equation with respect to the independent variable x to get

C'=2012500x2</

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