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MATH / Two runners, A and B, start at a point O on a linear track, and start running in the same direction. Runner B runs three times as fast as runner A. An observer stands at point P so…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Two runners, AA and B,B, start at a point OO on a linear track, and start running in the same direction. Runner BB runs three times as fast as runner A.A. An observer stands at point PP so that OP¯\overline{OP} is perpendicular to the track. Find the maximum of ∠APB,\angle APB, in degrees. [asy] unitsize(2 cm); pair A, B, O, P; A = (0.4,0); B = (1.2,0); O = (0,0); P = (0,1); draw((-0.5,0)--(2,0)); draw(O--P); draw(P--A); draw(P--B); label("AA", A, S); label("BB", B, S); label("OO", O, S); label("PP", P, N); [/asy]
Plain-text mathematical notation (without MathML)
Two runners, A and B, start at a point O on a linear track, and start running in the same direction.  Runner B runs three times as fast as runner A.  An observer stands at point P so that (OP)¯ is perpendicular to the track.  Find the maximum of ∠APB, in degrees.

[asy]
unitsize(2 cm);

pair A, B, O, P;

A = (0.4,0);
B = (1.2,0);
O = (0,0);
P = (0,1);

draw((-0.5,0)--(2,0));
draw(O--P);
draw(P--A);
draw(P--B);

label("A", A, S);
label("B", B, S);
label("O", O, S);
label("P", P, N);
[/asy]
Original LaTeX notation
Two runners, $A$ and $B,$ start at a point $O$ on a linear track, and start running in the same direction.  Runner $B$ runs three times as fast as runner $A.$  An observer stands at point $P$ so that $\overline{OP}$ is perpendicular to the track.  Find the maximum of $\angle APB,$ in degrees.

[asy]
unitsize(2 cm);

pair A, B, O, P;

A = (0.4,0);
B = (1.2,0);
O = (0,0);
P = (0,1);

draw((-0.5,0)--(2,0));
draw(O--P);
draw(P--A);
draw(P--B);

label("$A$", A, S);
label("$B$", B, S);
label("$O$", O, S);
label("$P$", P, N);
[/asy]

level

Level 3

type

Precalculus

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Source and history

task_id
e8528ad1-af97-55dc-8334-d8b96adc2a1e
task_key
precalculus--test--e8528ad1-af97-55dc-8334-d8b96adc2a1e
task_revision_id
1
config / split
precalculus / test

Official source

initial import