2008 AMC 12B Problem 3 — Solution | LIVE by Po-Shen Loh

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2008 AMC 12B Problem 3

Attempt Problem 3 of the 2008 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

3.

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?

$270{,}000$

$385{,}000$

$400{,}000$

$430{,}000$

$700{,}000$

Answer: C

Concepts:optimization extremal argument

Difficulty rating: 1100

Small Hint:

To make one salary as large as possible, pay the other $20$ players the minimum allowed

Big Hint:

The maximum is $\$700{,}000$ minus the total of $20$ minimum salaries

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.

Problem 3 in Other Years

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