A pollutant plume moves through ocean water

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

A pollutant plume moves through ocean water. Its center of mass is modeled by the vector function

\mathbf P(u)=u\,\langle \cos u,\,\sin u,\,u^2\rangle\text{ m},

where

u

represents salinity (dimensionless). Field measurements suggest salinity varies with time according to

u=g(t)=3e^{0.4t}-2

for

t\ge 0

hours. (a) Determine the time rate of change of salinity,

g'(t)

. (b) Find an explicit formula for the plume’s velocity vector

\mathbf V(t)=\dfrac{d\mathbf P}{dt}

. (c) Calculate the instantaneous speed of the plume at

t=1

hour. Give your answer to two decimal places and include units. (d) Explain, in complete sentences, how the scalar factor

u

\mathbf P(u)

influences both the direction and the magnitude of

\mathbf V(t)