Two skaters start from the same origin on a frozen lake at seconds

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Two skaters start from the same origin on a frozen lake at

t=0

seconds. Skater A moves along the

x

-axis according to

x_A(t)=12t-0.5t^2\quad (\text{meters}),

while Skater B moves along the

y

-axis according to

y_B(t)=8t\quad (\text{meters}).

For

t\ge 0

, let

S(t)=x_A(t)^2+y_B(t)^2

denote the square of the distance between the two skaters. (a) Compute

\dfrac{dS}{dt}

as a function of

t

. Simplify completely. (b) Solve

\dfrac{dS}{dt}=0

on the interval

0\le t\le 20

s. Identify all critical time(s) and classify them using the sign of

\dfrac{dS}{dt}

. (c) Calculate the minimal distance between the skaters during this interval. (d) Provide a brief explanation of how the sign analysis of

\dfrac{dS}{dt}

supports your answer in part (b).