A weight is attached to a spring that is oscillating up and down. It takes 6 sec. for the spring to compete one cycle, and the distance from the highest to lowest point is 7 in. Write an equation that models the position of the weight at time t seconds?

So we have a weight oscillating due to the action of a spring. We are told that its period (i.e. the time it takes it to complete a cycle) is 6 seconds and the diference between its two extreme positions is 7 in. This kind of movement can be modeled with a sinusoidal equation i.e. an expression with a sine. Let's have a look at the following function:

[tex]f(t)=A\sin(\frac{2\pi}{T}t)[/tex]

Where T is the period of the function and A is known as the amplitude. As we saw before, the period of our function must be equal to 6 because it takes the weight 6 seconds to complete a full cycle. Then we have:

[tex]f(t)=A\sin(\frac{2\pi}{6}t)=A\sin(\frac{\pi}{3}t)[/tex]

In order to find A let's take into account the behaviour of the sine. The maximum value of the sine is 1 and the minimum is -1. Then the extreme values of the function are A and -A. This are also the highest and lowest position of the weight. We are told that their difference must be equal to 7 in so we have the following equation for A:

[tex]\begin{gathered} A-(-A)=7 \\ A+A=7 \\ 2A=7 \end{gathered}[/tex]

We divide both sides by 2 and we obtain A:

[tex]\begin{gathered} \frac{2A}{2}=\frac{7}{2} \\ A=3.5 \end{gathered}[/tex]

Therefore we get:

[tex]f(t)=3.5\sin(\frac{\pi}{3}t)[/tex]

Answer

Then the answer is:

[tex]3.5\sin(\frac{\pi}{3}t)[/tex]