5. calculator allowed\nthe position of an object attached to a spring is given by ( y(t)=\frac{1}{6}cos(5t)-\…

Explanation:

Step1: Find the velocity function

Velocity (v(t)) is the derivative of position function (y(t)). Using the chain rule ((\cos(u))^\prime=-\sin(u)\cdot u^\prime) and ((\sin(u))^\prime=\cos(u)\cdot u^\prime) where (u = 5t) and (u^\prime=5). (y(t)=\frac{1}{6}\cos(5t)-\frac{1}{4}\sin(5t)) (v(t)=y^\prime(t)=\frac{1}{6}\times(-\sin(5t))\times5-\frac{1}{4}\times\cos(5t)\times5) (v(t)=-\frac{5}{6}\sin(5t)-\frac{5}{4}\cos(5t))

Step2: Set (v(t) = 0)

(-\frac{5}{6}\sin(5t)-\frac{5}{4}\cos(5t)=0) Divide both sides by (-\frac{5}{12}) (since (-\frac{5}{12}\neq0)): (2\sin(5t)+ 3\cos(5t)=0) (2\sin(5t)=-3\cos(5t)) (\tan(5t)=-\frac{3}{2})

Step3: Solve for (t) in the interval (t\in[0,4])

Let (x = 5t), then (x\in[0,20]) The general solution of (\tan(x)=-\frac{3}{2}) is (x = \arctan(-\frac{3}{2})+n\pi), (n\in\mathbb{Z}) (\arctan(-\frac{3}{2})\approx - 0.9828) (in radians), and using the property (\tan(x)=\tan(x + n\pi)) (x\approx- 0.9828 + n\pi) For (x\in[0,20]): When (n = 1), (x_1\approx-0.9828+\pi\approx2.1588) When (n = 2), (x_2\approx-0.9828 + 2\pi\approx5.3004) When (n = 3), (x_3\approx-0.9828+3\pi\approx8.4420) When (n = 4), (x_4\approx-0.9828 + 4\pi\approx11.5836) When (n = 5), (x_5\approx-0.9828+5\pi\approx14.7252) When (n = 6), (x_6\approx-0.9828+6\pi\approx17.8668) When (n = 7), (x_7\approx-0.9828+7\pi\approx21.0084>20)

Since (x = 5t), (t=\frac{x}{5}) The number of solutions for (t\in[0,4]) (i.e., (x\in[0,20])) is (6)

Answer:

D. Six