Question

A spring is oscillating vertically according to the equation y = 3cos(2t-pi), where y is the displacement in feet and t is time in seconds. At what time will thespring be 2 ft. high? Round your answer to the nearest tenth.

Answer 1

Answer: 25.7 seconds

Step-by-step explanation

Equation

`y=3\cos(2t-\pi)`

where y is the displacement in feet and t is time in seconds.

We are asked to find t when y = 2ft, meaning that we have to solve the equation for t:

0. Replacing the information given by setting the equation to 2:

`3\cos(2t-\pi)=2`

2. Dividing both sides of the equation against 3 and simplifying:

`(3\cos(2t-\pi))/(3)=(2)/(3)`
`\cos(2t-\pi)=(2)/(3)`

2. Applying secant (cos⁻¹) to both sides:

`\cos^(-1)(\cos(2t-\pi)=\cos^(-1)((2)/(3))`
`(2t-\pi)=\cos^(-1)((2)/(3))`

3. Adding π to both sides of the equation:

`2t-\pi+\pi=\cos^(-1)((2)/(3))+\pi`
`2t=\cos^(-1)((2)/(3))+\pi`

4. Dividing both sides of the equation against two:

`(2t)/(2)=(\cos^(-1)((2)/(3))+\pi)/(2)`
`t=(\cos^(-1)((2)/(3))+\pi)/(2)`

5. Simplifying we get:

`t\approx(51.33)/(2)=25.67`