Question

a mass on a spring bounces up and down in simple harmonic motion, modeled by the function ( s(t)=6sin t ) where ( s ) is measured in centimeters and ( t ) is measured in seconds. find the rate at which the spring is oscillating at ( t = 7 ) s. round your answer to four decimal places. ( mathrm{cm} / mathrm{s} ) question help: video message instructor submit question jump to answer

Explanation:

Step1: Find the derivative of \(s(t)\)

The derivative of \(y = a\sin t\) is \(y^\prime=a\cos t\). For \(s(t) = 6\sin t\), using the formula \(\frac{d}{dt}(\sin t)=\cos t\) and the constant - multiple rule \(\frac{d}{dt}(cf(t)) = c\frac{d}{dt}(f(t))\) (where \(c = 6\) and \(f(t)=\sin t\)), we get \(s^\prime(t)=6\cos t\).

Step2: Evaluate the derivative at \(t = 7\)

Substitute \(t = 7\) into \(s^\prime(t)\). So \(s^\prime(7)=6\cos(7)\). Using a calculator, \(\cos(7)\approx0.753902\) (in radians). Then \(s^\prime(7)=6\times0.753902 = 4.523412\approx4.5234\)

Answer:

\(4.5234\)