Track / Stage 5 / 141 of 400 #741 of 1964

Problem 741

AIME late

Algebra Difficulty 5.3 Find the answer

Two objects $A$ and $B$ are $343 \mathrm{~m}$ apart. Both start moving towards each other at the same moment with uniform acceleration. One object has an initial velocity of $3 \mathrm{~m}$ and an acceleration of $5 \mathrm{~m}$ ; the other object has an initial velocity of $4 \mathrm{~m}$ and an acceleration of $7 \mathrm{~m}$ . The question is, after how much time will the two objects meet, and at what distance from their starting points?

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

If two bodies meet after $t \mathrm{~s}$ , then the distance covered by $A$ is $3 t+\frac{5 t^{2}}{2}$ , and the distance covered by $B$ is $4 t+\frac{7 t^{2}}{2}$ ; thus

$3 t+\frac{5 t^{2}}{2}+4 t+\frac{7 t^{2}}{2}=343$

$6 t^{2}+7 t=343$

from which

$t=\frac{-7 \pm 91}{12}$

Since in this case $t$ can only be positive, therefore $t=7 \mathrm{~s}$ . During this time, $A$ covers $143.5 \mathrm{~m}$ , and $B$ covers $199.5 \mathrm{~m}$ .

Number of solutions: 27.

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