Jackson is conducting an experiment for his Physics class. He attaches a weight to the bottom of a metal spring. He then pulls t
he weight down so that it is a distance of six inches from its equilibrium position. Jackson then releases the weight and finds that it takes four seconds for the spring to complete one oscillation. Which of the following functions best models the position of the weight?
We have that the spring is going to have a sin or a cos equation. We have that the maximum distance of the spring is 6 inches and it is achieved at t=0. Let's fix this as the positive edge. Until now, we have that the function is of the form: 6sin(at+B). We have that the period is 4 minutes and hence that the time component in the equation needs to make a period (2pi) in 4 minutes. Thus 4min*a=2p, a=2p/4=pi/2. In general, a=2pi/T where a is this coefficient, T is the period. Finally, for B, since sin(pi/2)=1, we have that B=pi/2 because when t=0, we have that 6sin(B)=6. Substituting, we have f(t)=6sin(pi*t/2+pi/2)=6cos(pi*t/2) by trigonometric identities.
We will use u-substitute:Then for substitution:dx=u du. and integral becomes:=u-ln(u+1)=. Now we will change the values of limits: =4-ln(5)-2+ln(3)=2+ln(0,6)=2-0.51=1.49
The distance between the walkers increases at the rate of (8 mph) -(2 mph) = 6 mph. That distance will be 4.5 mi after ... (4.5 mi)/(6 mi/h) = 4.5/6 h = 3/4 h
Then for substitution:dx=u du. and integral becomes:
=u-ln(u+1)=
. Now we will change the values of limits: 
=4-ln(5)-2+ln(3)=2+ln(0,6)=2-0.51=1.49