<p>The pirate ship Shark is chasing down a ship Arica loaded with treasure. On a rectangular grid in units of kilometres at time $t=0$, Arica is at $\left( 0,0 \right)$ and Shark is at $\left( 10,-1 \right)$. Half an hour later, Arica is at $\left( 20,2 \right)$ and Shark is at $\left( 90,9 \right)$. Assume that both ships are following linear paths at constant speeds.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the Cartesian equation for the path of each ship.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find parametric equations that model the motion of each ship using $t$ as the parameter in hours.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Will Shark intercept Arica if neither ship changes course or speed?</span></div></li></ol>

The pirate ship Shark is chasing down a ship Arica loaded with treasure. On a rectangular grid in units of kilometres at time $t=0$, Arica is at $\left( 0,0 \right)$ and Shark is at $\left( 10,-1 \right)$. Half an hour later, Arica is at $\left( 20,2 \right)$ and Shark is at $\left( 90,9 \right)$. Assume that both ships are following linear paths at constant speeds.

Find the Cartesian equation for the path of each ship.

Find parametric equations that model the motion of each ship using $t$ as the parameter in hours.

Will Shark intercept Arica if neither ship changes course or speed?

Parametric Equations

Graphing Techniques

International Baccalaureate Higher Level Mathematics

IB HL Math

ACJC Parametric Equations Tutorial 4 Q7

**Arica:** At $t=0$: $(0,0)$, at $t=0.5$: $(20,2)$

Slope $m_A = \frac{2-0}{20-0} = \frac{1}{10}$

**Shark:** At $t=0$: $(10,-1)$, at $t=0.5$: $(90,9)$

Slope $m_S = \frac{9-(-1)}{90-10} = \frac{10}{80} = \frac{1}{8}$

$y = \frac{1}{8}x - \frac{90}{8} + 9 = \frac{1}{8}x - \frac{9}{4}$

**Shark:** $y = \frac{1}{8}x - \frac{9}{4}$

**Arica:** $D = 20$ km in $T = 0.5$ hours

$S_x = \frac{20}{0.5} = 40$ km/h, $S_y = \frac{2}{0.5} = 4$ km/h

**Shark:** $D_x = 80$ km, $D_y = 10$ km in $T = 0.5$ hours

$S_x = \frac{80}{0.5} = 160$ km/h, $S_y = \frac{10}{0.5} = 20$ km/h

**Arica:** $x_A = 40t$, $y_A = 4t$; $t \geq 0$

**Shark:** $x_S = 160t + 10$, $y_S = 20t - 1$; $t \geq 0$

$x_S = 160\left(\frac{1}{16}\right) + 10 = 20$

$x_A = 40\left(\frac{1}{16}\right) = 2.5$

Since when $y_A = y_S$, then $x_A \neq x_S$,

ACJC Parametric Equations Tutorial 4 Q7

ACJC Parametric Equations Tutorial 4 Q7

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