Question

Two sides of a triangle are 5 and 8 and the angle between them is increasing at .05 rad/sec. how fast is the distance between the tips of the sides increasing when the angle is pi/4

Answer 1

let that distance be x
angle be Ø

by the cosine law:
x^2 = 25+64-2(5)(8)cosØ
x^2 = 89 - 80cosØ

2x dx/dt = 80sinØ dØ/dt
dx/dt = 40sinØ dØ/dt / x ***

when Ø = π/4 or 45°
x^2 = 89-80sin π/4
= 32.431...
x = 5.6948... (I stored that in memory)

back in ***

dx/dt = 40(sin (π/4)) (-.05)/5.6948...
= -.24833.. units per second

(the - signs shows that the distance is decreasing)