2008 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2008 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 12B solutions, or check the answer key.

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Concepts:optimization extremal argument

Difficulty rating: 1100

A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $\$15{,}000,$ and that the total of all players' salaries for each team cannot exceed $\$700{,}000.$ What is the maximum possible salary, in dollars, for a single player?

$270{,}000$

$385{,}000$

$400{,}000$

$430{,}000$

$700{,}000$

Solution:

One player earns the most when the other $20$ players each receive the minimum salary of $\$15{,}000.$

Thus the maximum salary is $\$700{,}000 - 20 \cdot \$15{,}000 = \$700{,}000 - \$300{,}000 = \$400{,}000.$

Thus, the correct answer is C.

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