MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
A logarithmic spiral is modeled by r=θ for 0<θ≤4π. A fixed observation station is located at the point (5,0) in the Cartesian plane. A particle travels along the spiral from the pole outward.
(a) Express the square of the distance D between the moving particle (x,y) and the station (5,0) as a function of θ. Use only algebraic manipulation and trigonometric identities; do not simplify numerically.
(b) Determine every value of θ in the interval (0,4π] at which this distance is minimized. Show all analytical steps leading to your conclusion.
(c) Verify that the value(s) of θ found in part (b) correspond to local minima rather than maxima.
(d) Compute the minimal distance and interpret its physical meaning for the particle–station system.