2008 AMC 10B Problem 4

Below is the professionally curated solution for Problem 4 of the 2008 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 10B solutions, or check the answer key.

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Concepts:inequality optimization extremal argument

Difficulty rating: 840

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's salary is largest when the other $20$ players each earn the minimum $\$15{,}000.$

That leaves $\$700{,}000-20\cdot\$15{,}000=\$700{,}000-\$300{,}000=\$400{,}000$ for the single player.

Thus, the correct answer is C.

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