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
Name:
Anonymous2008-12-13 15:11
Too lazy to do pictures
a) Because things should always be in radians unless you're in third fucking grade, rewrite the equation as
sin x = 9/10 x
where x is in radians
The triangle defined by the given chord and the center of the circle splits into two right triangles with sides R sin x, R cos x, R. We are given R sin x = 135. But also the length of the arc is 2xR = 300 or xR = 150. Eliminate R and the formula you want drops out.
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
OP here ty ty - as for response 2 - i get the cosine rule usuage but what's Taylor expanding and bisection search?
NEW questions:
Let sec x = b where x - [pi/2, pi]. Find, in terms of x, two values of x in the range [-pi, pi] which satisfy each of the following equations:
a. sec x = -b
b. cosec x = b
normally can do these questions but this one makes it really confusing as the quadrant SHOULD be positive but it's negative.
Let tan x = b where x - [pi, 3pi/2]. Find, in terms of x, two values of x in the range [0, 2pi] which satisfy each of the following equations:
a. sec x = -c
b. cot x = c
Answers in the book (incase they're wrong) are:
1st q: a. pi - x, x - pi
b. pi/2 - x, x - 3pi/2
2nd q: a. 2pi - x, 3pi - x (3pi!?)
b. 3pi/2 - x, 5pi/2 - x
I got the first part of the first question but have no idea what method the answers are using
>>5
They're algorithms in numerical analysis to approximate the solution of an equation. It sounds like a crappy question to ask if you're not taught to do numerical analysis, unless they just want a crappy result you pull off a computer or from grinding your calculator. I also forgot about fixed point iteration: (radians) 10sin(xn)/9 = xn+1, and x∞ is the solution.
Name:
Anonymous2008-12-14 17:23
OP:ye, methinks they just wanted crappy results i pulled off from a calculator.