Trigonometric Functions & Equations

Question

A small windmill has its center 6 𝑚 above the ground and blades 2 𝑚 in length. In a steady wind, a point 𝑃 at the tip of one blade makes a complete rotation in 12 𝑠.

a) If the rotation begins at the highest possible point, draw a sketch showing 2 cycles. Clearly label your sketch.

Determine a function (an equation) that gives the height of point 𝑃 above the ground at time

What is the height at point P at 5s?

At what time during the first cycle is point P exactly 7 m above the ground?

Answer 1

ω = 2π/ Τ

That is ω = π/6

Then H = 6 +2 cos[( π/6) t], where H is the height above the ground at time t.

At time t=5, H = 6 +2 cos(5π /6)

H = 6 +2 cos(π-π/6)

H = 6 + 2 [ -cos(π/6) ]

H = 6 -2cos(π/6)

H =6 - 2√3/2

H = 6- √3 ≈ 4.268 m.

7 = 6 +2 cos[( π/6) t]

cos[( π/6) t] =1/2

cos[( π/6) t] = cos ( π/3) ⇒ πt/6 = π/3 ⇒ t = 2