A spring is hanging from a ceiling. The length L(t) (in cm ) of the spring as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a*sin(b*t)+d. At t=0, when the spring is exactly in the middle of its oscillation, its length is 7cm. After 0.5 seconds the spring reaches its maximum length, which is 12cm. Find L(t). t should be in radians. L(t)=

Question

A spring is hanging from a ceiling. The length  L(t) (in  cm ) of the spring as a function of time  t (in seconds) can be modeled by a sinusoidal expression of the form  a*sin(b*t)+d. At  t=0, when the spring is exactly in the middle of its oscillation, its length is  7cm. After 0.5 seconds the spring reaches its maximum length, which is  12cm. Find  L(t).  t should be in radians.  L(t)=

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Calculate Amplitude: The spring reaches its maximum length after 0.5 seconds, which is 12cm. The amplitude a is the difference between the maximum length and the equilibrium position, so a=12-7=5cm.Find Period: Since the spring reaches its maximum length at 0.5 seconds, we can find b by using the period of the sine function. The period T is the time it takes to complete one full cycle, and since the maximum occurs at 0.5 seconds, T=4*0.5=2 seconds. The formula for the period is T=(2*pi)/b, so b=(2*pi)/T.Calculate b: Calculating b using T=2 seconds, we get b=(2*pi)/2=pi.Sinusoidal Function Parameters: Now we have all the parameters for the sinusoidal function: a=5, b=pi, and d=7. The function L(t) is L(t)=a*sin(b*t)+d.Substitute Values: Substituting the values into L(t), we get L(t)=5*sin(pi*t)+7.

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