Question

if I have a board exactly 240 inches, how much is the rise in arc bowing the board to bring linear length to 238 inches?Ask

Answer 1

Answer on question 40457 – Math – Geometry

if I have a board exactly 240 inches, how much is the rise in arc bowing the board to bring linear length to 238 inches?

Solution

The length of the arc AB L is 240, the length of the chord l is 238. We know that

$$L = R \varphi , \qquad l = 2 R \sin {\frac {\varphi}{2}},$$

Where R is the radius of the circle and $\varphi$ is measured in radians. We get the system of equations

\left\{ \begin{array}{c} R \varphi = 240 \\ 2 R \sin {\frac {\varphi}{2}} = 238 \end{array} \right. \qquad \varphi \approx 2.02 \sin {\frac {\varphi}{2}} \\ \varphi \approx 0.488 \approx 29{}^{\circ}. \end{array}

Answer: $\approx 0.488 \approx 29{}^{\circ}$

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