Answer to Question #339970 in Trigonometry for sowmya

Question

1. Find the measure, to the nearest degree, of the acute central angle in the figure below.

(The radius is 21 and the arc length is 18.)

2 Explain what would happen to the measure of the central angle in the figure aboce if the radius were doubled (and the arc length remained unchanged).

Answer 1

(arc length) ÷ circumference = (central angle) ÷ 360°

arc length - L=18

Radius r=21

Circumference - C.

$D=2r=C/\pi$

$C=2r\pi=2(21)3.14=131.88$

$L/C=\alpha/360$

$18/131.88=\alpha/360$

$\alpha=360L/C=360L/2r\pi$

$\alpha=49.1 °$

if the radius were doubled (and the arc length remained unchanged C will be doubled, then central angle will become smaller for two times.

If R=2r, then C=2x2rπ=4rπ, then $\alpha_{new}=360L/4r\pi$

$\alpha/\alpha_{new}=360L(4r\pi)/(2r\pi(360L))=2$