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MATH / A circular sector has fixed perimeter, but the central angle θ and radius r are allowed to vary. [asy] unitsize(2 cm); pair A, B, O; A = (1,0); B = dir(110); O = (0,0); draw(A--O--…

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

problem

A circular sector has fixed perimeter, but the central angle θ\theta and radius rr are allowed to vary. [asy] unitsize(2 cm); pair A, B, O; A = (1,0); B = dir(110); O = (0,0); draw(A--O--B); draw(arc(O,1,0,110)); label("rr", (O + A)/2, S); label("rr", (O + B)/2, SW); label("θ\theta", (0.1,0.15)); [/asy] Which central angle θ\theta maximizes the area of the circular sector? Give the answer in radians. Note: The arc length of the sector is rθ,r \theta, and the area of the sector is r2θ2.\frac{r^2 \theta}{2}.
Plain-text mathematical notation (without MathML)
A circular sector has fixed perimeter, but the central angle θ and radius r are allowed to vary.

[asy]
unitsize(2 cm);

pair A, B, O;

A = (1,0);
B = dir(110);
O = (0,0);

draw(A--O--B);
draw(arc(O,1,0,110));

label("r", (O + A)/2, S);
label("r", (O + B)/2, SW);
label("θ", (0.1,0.15));
[/asy]

Which central angle θ maximizes the area of the circular sector?  Give the answer in radians.

Note: The arc length of the sector is rθ, and the area of the sector is (r²θ)/(2).
Original LaTeX notation
A circular sector has fixed perimeter, but the central angle $\theta$ and radius $r$ are allowed to vary.

[asy]
unitsize(2 cm);

pair A, B, O;

A = (1,0);
B = dir(110);
O = (0,0);

draw(A--O--B);
draw(arc(O,1,0,110));

label("$r$", (O + A)/2, S);
label("$r$", (O + B)/2, SW);
label("$\theta$", (0.1,0.15));
[/asy]

Which central angle $\theta$ maximizes the area of the circular sector?  Give the answer in radians.

Note: The arc length of the sector is $r \theta,$ and the area of the sector is $\frac{r^2 \theta}{2}.$

level

Level 4

type

Intermediate Algebra

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Source and history

task_id
a17d7ffc-f528-5b9a-95c4-f0bf1b602139
task_key
intermediate~5falgebra--test--a17d7ffc-f528-5b9a-95c4-f0bf1b602139
task_revision_id
1
config / split
intermediate_algebra / test

Official source

initial import