Track / Stage 5 / 249 of 400 #849 of 2000

Problem 849

AIME late

Algebra Difficulty 5.6 Find the answer

Robinson had 200,000 strands of hair when he was stranded on the deserted island. At that time, his hair strands were $5 \mathrm{~cm}$ long. The hair strands grew by $0.5 \mathrm{~mm}$ per day, but Robinson did not cut his hair because, on the one hand, he did not have the appropriate tools for it, and on the other hand, 50 strands of his hair fell out every day, which unfortunately did not regenerate.

How many days after being stranded did Robinson reach the maximum total length of his hair strands?

Your answer

Official solution

After $x$ days, Robinson had $200000 - 50 x$ strands of hair, each with a length of $5 + 0.05 x \, \text{cm}$ . The total length of the hair strands at that time is

$H = (200000 - 50 x)(5 + 0.05 x)$

After rearranging:

$H = -2.5 x^2 + 9750 x + 10^6 \, (\text{cm})$

$H$ is a quadratic expression in $x$ , which has a maximum because the coefficient of $x^2$ is negative. By the same transformation, we get:

$H = -2.5(x - 1950)^2 + 10506250$

From this, it is clear that $H$ is maximized when $x = 1950$ .

Therefore, the total length of the hair strands on Robinson's head reached its maximum after 1950 days. The maximum value is $10506250 \, \text{cm}$ , or approximately $105 \, \text{km}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.