Trig functions Math

Question

A ladder 16 ft long rests against a vertical wall. Let 𝜃 be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away

from the wall, how fast (in ft/rad) does x change with respect to 𝜃 when 𝜃 = pi/3?

ft/rad

A particle is moving with the given data. Find the position of the particle,

a(t) = 6 cos(t) − 7 sin(t), s(0) = 0, v(0) = 8

Answer 1

a) x = 16sinθ ft; dx/dθ = 16cosθ = 16cosπ/3 = 8ft/rad

b) Velocity v(t) = ∫(6cos(t) - 7sin(t))dt = 6sin(t) + 7cos(t) + C; v(0) = 7 + C = 8; C = 1

v(t) = 6sin(t) + 7cos(t) + 1;

Position s(t) = ∫(6sit(t) + 7cos(t) + 1)dt = - 6cos(t) + 7sin(t) + t + D; s(0) = - 6 + D = 0; D = 6.

s(t) = - 6cos(t) + 7sin(t) + t + 6