A curve on a light rail track is an arc of a circle of length 300m and the straight line joining the two ends of the curve is 270m long.
a) show that, if the arc subtends an angle of 2x(degrees) at the center of the circle, x is a solution of the equations
sin x(degrees) = (pi*x)/200
b) solve, correct to 2 d.p., the equation for x
a)
start with geometry, radians
r(2x) = 300
rsqrt(r^2 + r^2 - 2r^2cos(2x)) = 270
after algebra,
sin(x) = 9x/10 -> convert to degrees
b)
i) use electronic equation solver
ii) try some numbers, figure out x ~ 45 degrees, Taylor expand around 45, solve quadratic (more if not accurate enough)
iii) try some numbers, figure out that LHS > RHS for x << 45 degrees, and LHS < RHS for x = 90 >> 45 degrees; therefore, there is a root somewhere inside -> bisection search
iv) brute force numbers on your calculator