The Titanic gives out a distress signal giving its location by the grid reference (-50,175). A freighter at (100,-400) and the steamship the Carpathia at (225,100) hear and respond... The Titanic gives out a distress signal giving its location by the grid reference (-50,175). A freighter at (100,-400) and the steamship the Carpathia at (225,100) hear and respond. If the freight travels twice as fast as the Carpathia, which ship arrives first? [6 marks] Read more

Steps to Solve

1. Identify the coordinates of the ships and the Titanic

The coordinates are as follows:

• Titanic: $(-50, 175)$
• Freighter: $(100, -400)$
• Carpathia: $(225, 100)$

2. Calculate the distance from each ship to the Titanic using the distance formula

The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

• Distance from the Freighter to the Titanic:

  Let $(x_1, y_1) = (100, -400)$ and $(x_2, y_2) = (-50, 175)$

  $$ d_{freighter} = \sqrt{((-50 - 100)^2 + (175 - (-400))^2)} $$

  Simplifying further, we get:

  $$ d_{freighter} = \sqrt{(-150)^2 + (575)^2} $$

  $$ d_{freighter} = \sqrt{22500 + 330625} = \sqrt{352125} $$

• Distance from the Carpathia to the Titanic:

  Let $(x_1, y_1) = (225, 100)$ and $(x_2, y_2) = (-50, 175)$

  $$ d_{carpathia} = \sqrt{((-50 - 225)^2 + (175 - 100)^2)} $$

  Simplifying further, we get:

  $$ d_{carpathia} = \sqrt{(-275)^2 + (75)^2} $$

  $$ d_{carpathia} = \sqrt{75625 + 5625} = \sqrt{81250} $$

3. Calculate the numerical values of the distances

• Calculate ( d_{freighter} ) and ( d_{carpathia} ):

$$ d_{freighter} = \sqrt{352125} \approx 594.51 $$

$$ d_{carpathia} = \sqrt{81250} \approx 285.52 $$

4. Determine the travel speeds and times

Let ( v ) be the speed of the Carpathia. Then the freighter's speed is ( 2v ).

• Time taken by the Carpathia:

$$ t_{carpathia} = \frac{d_{carpathia}}{v} $$

• Time taken by the Freighter:

$$ t_{freighter} = \frac{d_{freighter}}{2v} $$

5. Set up the inequality to compare arrival times

To find out which ship arrives first, set up the inequality:

$$ t_{carpathia} < t_{freighter} $$

Substituting the distances into the inequality:

$$ \frac{d_{carpathia}}{v} < \frac{d_{freighter}}{2v} $$

This simplifies to:

$$ 2d_{carpathia} < d_{freighter} $$

6. Determine which ship arrives first

Using the calculated distances:

$$ 2 \times 285.52 < 594.51 $$

Calculate:

$$ 571.04 < 594.51 $$

This is true, meaning the Carpathia arrives first.