Examples Hyperbolic functions are widely used in engineering. The equation for a catenary, the shape of a rope suspended from two ends, is f(x) = a cosh(x/a) - a if the ends are x = - b and x = + b at equal heights and the lowest point is f(x) = 0. Suppose the constant is a = 10. The value x = 0 is where the rope hangs lowest, since it reaches its minimum where f'(x) = 0 and if f'(x) = 10 sinh(x/10) [1/10] = sinh(x/10) = 0 we need sinh(x) = 0. We can calculate how much lower the middle is than the ends using f(b) = 10 cosh(b/10) - 10, so if b = 5, the height of each end is f(5) = 1.28.[1]
The equation for the velocity in meters per second of an idealized surface wave travelling across a lake is
where g = 9.8 m/s is gravity's acceleration, is the wavelength from crest to crest in meters, and d is the depth of undisturbed water in meters.[2] Thus, if the wavelength is 12 meters and the velocity is 4 meters per second, we can find the depth of the lake from
which implies that
- ^ Adapted from | "The Hanging Cable Problem for Practical Applications," Neil Chatterjee & Bogdan G. Nita, Atlantic Electronic Journal of Mathematics, 4(1): 70-77 (Winter 2010).
- ^ Calculus: Early Transcendentals, 2nd Edition, by William Briggs, Lyle Cochran, Bernard Gillett & Eric Schulz, p. 501.