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 $20.00 to prepare, and it costs $125.00 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

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ajeigbeibraheem ajeigbeibraheem · Answer 1 · 2021-06-24T04:57:03+02:00

Answer:

25

Step-by-step explanation:

From the given information;

Numbers of posters that can be printed in an hour = no of impression/hour × no of plate utilized in each impression.

= 1000x

Thus, the required number of hours it will take can be computed as:

[tex]\implies \dfrac{100000}{1000x} \\ \\ =\dfrac{100}{x}[/tex]

cost per hour = 125

If each plate costs $20 to make, then the total number of plate will equal to 40x

∴

The total cost can be computed as:

[tex]C(x) = (\dfrac{100}{x}) \times 125 + 20 x --- (1)[/tex]

[tex]C'(x) = (-\dfrac{12500}{x^2}) + 20 --- (2)[/tex]

At C'(x) = 0

[tex]\dfrac{12500}{x^2} = 20[/tex]

[tex]\dfrac{12500}{20} = x^2[/tex]

[tex]x^2= 625[/tex]

[tex]x = \sqrt{625}[/tex]

x = 25

[tex]C'' (x) = -12500 \times \dfrac{-2}{x^3} +0[/tex]

[tex]C'' (x) = \dfrac{25000}{x^3}[/tex]

where; x = 25

[tex]C'' (x) = \dfrac{25000}{25^3}[/tex]

C''(x) = 1.6

Thus, at x = 25, C'' > 0

As such, to minimize the cost, the printer needs to make 25 metal plates.