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Let's say that Sally is playing a special game of tag where she has to run to a safe space to win. The safe space is 157 feet away, and she can run 10 feet in one second. keep in mind this is hypothetical. Simon, the chaser, is 75 feet away from Sally, and he can run twice as fast. who will win the game?

7 answers

To determine who will win the game, we can calculate how long it will take both Sally and Simon to reach the safe space.

For Sally:

  • Distance to safe space = 157 feet
  • Speed = 10 feet/second
  • Time = Distance / Speed = 157 feet / 10 feet/second = 15.7 seconds

For Simon:

  • Distance to safe space = 157 feet
  • Speed = 20 feet/second (twice Sally's speed)
  • Time = Distance / Speed = 157 feet / 20 feet/second = 7.85 seconds

Now we compare the times:

  • Sally takes 15.7 seconds.
  • Simon takes 7.85 seconds.

Since Simon can reach the safe space in 7.85 seconds, while Sally takes 15.7 seconds, Simon will win the game.

great, now assume that simon is not running to the safe spot, but is running after sally. will simon reach sally first, or will sally be safe

In this scenario, we need to determine if Simon will catch up to Sally before she reaches the safe space.

  1. Sally's Situation:

    • Distance to safe space = 157 feet
    • Speed = 10 feet/second
    • Time for Sally to reach safe space = \( \frac{157 \text{ feet}}{10 \text{ feet/second}} = 15.7 \text{ seconds} \)

  2. Simon's Situation:

    • Distance he has to cover to reach Sally = 75 feet (initial distance)
    • Speed = 20 feet/second
    • Time for Simon to reach Sally = \( \frac{75 \text{ feet}}{20 \text{ feet/second}} = 3.75 \text{ seconds} \)

Now we compare the times:

  • Sally takes 15.7 seconds to reach the safe space.
  • Simon takes only 3.75 seconds to catch up to Sally.

Since Simon reaches Sally in 3.75 seconds, which is well before Sally reaches the safe space (15.7 seconds), Simon will catch Sally before she can get to the safe spot, meaning Sally will not be safe.

how fast would sally have to run to be able to reach the safe place before simon reaches her?

To determine how fast Sally would need to run in order to reach the safe space before Simon catches her, we can start by noting the time it takes Simon to reach Sally.

From the previous calculation, Simon takes 3.75 seconds to reach Sally. We want Sally to reach the safe space in less time than that.

  1. Safe Space Distance:

    • Distance to the safe space = 157 feet

  2. Time for Sally to reach the safe space (must be less than Simon's time to catch her):

    • Time = 3.75 seconds (Simon’s time)

  3. Sally's Required Speed: We can use the formula for speed: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] To find the speed that allows Sally to reach the safe space in 3.75 seconds: \[ \text{Speed} = \frac{157 \text{ feet}}{3.75 \text{ seconds}} \approx 41.87 \text{ feet/second} \]

Thus, Sally would need to run approximately 41.87 feet/second to reach the safe space before Simon can catch her.

Two-part question.

Part 1: if the safe spot were 200 feet away and sally could run 50 feet in one second, how far much faster would simon have to be?

now, let's put sally back to 10 feet per second, the safe spot Is 157 feet away. now, how far away would Simon have to be for sally to win at 10 feet per second?

Let's tackle each part of your question step by step.

Part 1:

If the safe spot is 200 feet away and Sally can run 50 feet in one second, how much faster would Simon have to be?

  1. Sally's time to reach the safe spot:

    • Distance = 200 feet
    • Speed = 50 feet/second
    • Time = Distance / Speed = \( \frac{200 \text{ feet}}{50 \text{ feet/second}} = 4 \text{ seconds} \)

  2. Simon's required speed to reach Sally (who starts 75 feet away):

    • Distance (to Sally) = 75 feet
    • To catch Sally in 4 seconds, Simon needs to reach her in that same time.
    • His speed, if he covers 75 feet in 4 seconds, would be: \[ \text{Speed} = \frac{75 \text{ feet}}{4 \text{ seconds}} = 18.75 \text{ feet/second} \]

  3. Sally running at 50 feet/second needs Simon to be faster than 18.75 feet/second. If Simon is currently running at 20 feet/second, he would need to run at least \( 18.75 \text{ feet/second} \) to have a chance.

Part 2:

If Sally runs at 10 feet per second and the safe spot is 157 feet away, how far away would Simon have to be for Sally to win?

  1. Sally's time to reach the safe space:

    • Distance = 157 feet
    • Speed = 10 feet/second
    • Time = \( \frac{157 \text{ feet}}{10 \text{ feet/second}} = 15.7 \text{ seconds} \)

  2. Simon's speed remains at 20 feet/second. We need to calculate the maximum distance from which Simon can start such that he takes longer than 15.7 seconds to reach Sally.

    • Time Simon has to take = 15.7 seconds
    • Distance Simon can cover in that time: \[ \text{Distance} = \text{Speed} \times \text{Time} = 20 \text{ feet/second} \times 15.7 \text{ seconds} = 314 \text{ feet} \]

  3. Simon can start up to 314 feet away from Sally. Since Sally is starting from the safe spot (157 feet away), we want to find out how far away Simon can start from her initial position:

    • The maximum distance from where Simon can start is 314 feet from Sally’s starting position (at 0 feet).
    • Thus, if Sally is 157 feet away from the safe space, Simon can start \( 314 - 157 = 157 \) feet away from her.

So, for Sally to win, Simon can be at most 157 feet away from her when she starts running towards the safe space.