the motion of a weight that hangs from a spring is represented by the equation (h = - 5sin(\frac{3pi}{4}t))…

the motion of a weight that hangs from a spring is represented by the equation (h = - 5sin(\frac{3pi}{4}t)). it models the weights height, (h), in inches above or below the rest position as a function of time, (t), in seconds. approximately when will the object be 4 inches below the rest position? round to the nearest hundredth, if necessary.\n0 seconds\n0.29 seconds\n0.39 seconds\n1.95 seconds

the motion of a weight that hangs from a spring is represented by the equation (h = - 5sin(\frac{3pi}{4}t)). it models the weights height, (h), in inches above or below the rest position as a function of time, (t), in seconds. approximately when will the object be 4 inches below the rest position? round to the nearest hundredth, if necessary.\n0 seconds\n0.29 seconds\n0.39 seconds\n1.95 seconds

Answer

Explanation:

Step1: Set up the equation

We are given $h = - 5\sin(\frac{3\pi}{4}t)$ and we want to find $t$ when $h=-4$. So we set up the equation $-4=-5\sin(\frac{3\pi}{4}t)$.

Step2: Solve for $\sin(\frac{3\pi}{4}t)$

Divide both sides of the equation by - 5. We get $\sin(\frac{3\pi}{4}t)=\frac{4}{5}=0.8$.

Step3: Use the inverse - sine function

We know that if $\sin(x) = a$, then $x=\sin^{-1}(a)+2k\pi$ or $x=\pi-\sin^{-1}(a)+2k\pi$, $k\in\mathbb{Z}$. For $\sin(\frac{3\pi}{4}t)=0.8$, we first find $\sin^{-1}(0.8)\approx0.9273$. So $\frac{3\pi}{4}t\approx0.9273 + 2k\pi$ or $\frac{3\pi}{4}t\approx\pi - 0.9273+2k\pi$. We are looking for the first non - negative solution, so we start with $k = 0$. For $\frac{3\pi}{4}t\approx0.9273$, we solve for $t$: $t=\frac{0.9273\times4}{3\pi}\approx\frac{3.7092}{9.4248}\approx0.39$. For $\frac{3\pi}{4}t\approx\pi - 0.9273\approx3.1416 - 0.9273 = 2.2143$, $t=\frac{2.2143\times4}{3\pi}\approx\frac{8.8572}{9.4248}\approx0.94$. The first non - negative solution is the one we want.

Answer:

0.39 seconds