answer:Let's evaluate each part using the provided information. (a) Displacement at t = 1.7 s: {eq}begin{align*} x &= (4.1 m) cosleft[left(3pi rad/sright)times 1.7 s + frac{pi}{3} radright] x &= -0.85 m end{align*}{/eq} (b) Velocity at t = 1.7 s: {eq}begin{align*} v &= -omega x v &= -(3pi rad/s) times (-0.85 m) v &= 8.03 m/s end{align*}{/eq} (c) Acceleration at t = 1.7 s: {eq}begin{align*} a &= -omega^2 x a &= -(3pi rad/s)^2 times (-0.85 m) a &= 75.5 m/s^2 end{align*}{/eq} (d) Phase of the motion: {eq}begin{align*} phi &= (3pi rad/s) times 1.7 s + frac{pi}{3} rad phi &= 17.07 rad end{align*}{/eq} (e) Frequency and period of the motion: From the standard equation of SHM, {eq}x = A cos(omega t + Phi_0){/eq}, we have {eq}omega = 3pi rad/s{/eq}. Frequency, {eq}f = frac{omega}{2pi}{/eq}: {eq}begin{align*} f &= frac{3pi rad/s}{2pi} f &= 1.5 Hz end{align*}{/eq} Period, T = {eq}frac{2pi}{omega}{/eq}: {eq}begin{align*} T &= frac{2pi}{3pi rad/s} T &= 0.67 s end{align*}{/eq} In summary, the answers are: (a) -0.85 m (b) 8.03 m/s (c) 75.5 m/s^2 (d) 17.07 rad (e) Frequency: 1.5 Hz, Period: 0.67 s

answer:The given series is a geometric series with the first term a = -frac{59}{87} and the common ratio r = frac{1}{5}. To find the sum of an infinite geometric series, we use the formula: [ S = frac{a}{1 - r} ] Substituting the values, we get: [ S = frac{-frac{59}{87}}{1 - frac{1}{5}} ] [ S = frac{-frac{59}{87}}{frac{4}{5}} ] [ S = -frac{59}{87} times frac{5}{4} ] [ S = -frac{295}{348} ] Therefore, the sum of the infinite geometric series is -frac{295}{348}.

answer:Let's fix the "unwanted" dancer at the north position. The adjacent positions can be filled in 4cdot3 ways by musicians. The remaining musicians and dancers can be placed in 3!5! ways, maintaining the alternating arrangement. Therefore, the number of arrangements is 4cdot3cdot5!3! = 8640.

answer:To find the components of the boat's velocity, we can use trigonometry. The x-component is the velocity multiplied by the cosine of the angle between the velocity vector and the x-axis, and the y-component is the velocity multiplied by the sine of the angle between the velocity vector and the y-axis. In this case, the velocity is 20 mph and the angle is 45 degrees. Therefore, the x-component is 20 cos(45) mph = 14.14 mph, and the y-component is 20 sin(45) mph = 14.14 mph. The x-component represents the boat's velocity in the eastward direction, and the y-component represents the boat's velocity in the northward direction. The x-component of the boat's velocity is 20 cos(45) mph, and the y-component is 20 sin(45) mph.