A particle moves in the plane so that its position at time (seconds) is given by the…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A particle moves in the plane so that its position at time

t

(seconds) is given by the vector function

\mathbf{r}(t)=\langle t^{2},\,k t^{3}\rangle

0\le t\le2

, where

k>0

is a constant. Over this time interval, the particle travels a total distance of

10\,\text{m}

. (a) Express the distance traveled,

D(k)

, as a definite integral in terms of

k

. (b) Evaluate the integral obtained in part (a) to obtain a simplified algebraic formula for

D(k)

. (c) Determine the value of

k

(to three decimal places) that yields a total distance of

10\,\text{m}

. (d) The particle’s speed increases with time. Explain how the parameter

k

influences the rate at which speed grows, supporting your answer with calculus-based reasoning.