Hugo's bike tire has a piece of gum stuck to it. The distance G(t) (in cm ) between the gum and the sidewalk 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, the gum is halfway between the ground and its maximum height, at 35cm. The gum reaches its maximum height of 70cm from the ground (pi)/(20) seconds later. Find G(t). t should be in radians.

Question

Hugo's bike tire has a piece of gum stuck to it. The distance  G(t) (in  cm ) between the gum and the sidewalk 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, the gum is halfway between the ground and its maximum height, at  35cm. The gum reaches its maximum height of  70cm from the ground  (pi)/(20) seconds later. Find  G(t).  t should be in radians.

Bytelearn · Accepted Answer

Calculate Midline: At t=0, the gum is halfway to its maximum height, which is 70 cm, so the midline d is 35 cm.Determine Amplitude: The amplitude a is the distance from the midline to the maximum height, so a = 70 cm - 35 cm = 35 cm.Find Period: The gum reaches its maximum height at (pi)/(20) seconds, which is a quarter of the period of the sinusoidal function. Therefore, the period T is 4 * (pi)/(20) = (pi)/5 seconds.Calculate b Value: The value of b is related to the period by the formula b = 2*pi/T. So b = 2*pi/(pi/5) = 10.Determine Phase Shift: Since the gum starts halfway between the ground and its maximum height, the sinusoidal function must be a cosine function shifted by pi/2 to the right. However, we can use the sine function with a phase shift to the left by pi/2. So the phase shift c is -pi/2.Final Sinusoidal Function: The function G(t) is then G(t) = a*sin(b*t + c) + d. Plugging in the values we get G(t) = 35*sin(10*t - pi/2) + 35.

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