this will probably be hard to visualize, but any help is appreciated. please be

Question

this will probably be hard to visualize, but any help is appreciated. please be AS DETAILED AS POSSIBLE IN YOUR EXPLANATION. SHOW ALLLL STEPS! SHOW ALLLLL WORK! . Problem: the figure above shows a spotlight shining on point P(x,y) on the shoreline of Crescent Island. The spotlight is located at the origin and is rotating. The portion of the shoreline on which the spotlight shines is in the shape of the parabola y = x^2 from the point (1,1) to point (5, 25). Let @ (@ = theta) be the angle between the beam of the light and the positive x-axis. . A) for what values of @ (theta) between 0 and 2pi does the spotlight shine on the shoreline? . B) find the x- and y- coordinates of point P in terms of tan @ (theta) . C) If the spotlight is rotating at the rate of one revolution per minute, how fast is the point P traveling along the shoreline at the instant it is at the point (3,9)?

Explanation / Answer

a) minimum @ = arctan(1/1) = pi/4 rads maximum @ = arctan(25/5) = 1.37 rads b) y/x = tan(@) x^2/x = x = tan(@) x = tan(@) y = x^2 y = tan^2(@) coordinates of point P = (tan(@), tan^2(@)) c) dz/dt = sqrt((dx/dt)^2 + (dy/dt)^2) x =tan(@) dx/dt = sec^2(@)d@/dt y = tan^2(@) dy/dt = 2tan(@)sec^2(@)d@/dt dz/dt = sqrt((sec^2(@)d@/dt)^2 + (2tan(@)sec^2(@)d@/dt)^2) d@/dt = 2pi rads/min tan(@) = 9/3 = 3 sec^2(@) = (3sqrt(10)/3)^2 = 10 dz/dt = sqrt((10*2pi)^2 + (2*3*10*2pi)^2) dz/dt = 121.66pi units/min dz/dt = 382.19 units/min

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