a mass on a spring bounces up and down in simple harmonic motion, modeled by the function ( s(t)=6sin t )…

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

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

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)