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3. A ball tossed to a height of 4 meters rebounds to 40% of its previous height. Find the distance the ball has traveled when it strikes the ground for the fifth time.

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To find the total distance traveled by the ball when it strikes the ground for the fifth time, we can analyze its motion as follows:

  1. First toss: The ball is tossed to a height of 4 meters.

  2. Rebound to 40%: After reaching the height of 4 meters, it rebounds to 40% of that height, which is: \[ 0.4 \times 4 = 1.6 \text{ meters} \]

  3. Motion sequence:

    • The ball travels up 4 meters (when tossed).
    • It then falls down 4 meters (to the ground).
    • It rebounds up 1.6 meters.
    • It falls down 1.6 meters.
    • Then it will rebound to: \[ 0.4 \times 1.6 = 0.64 \text{ meters} \]
    • It falls down 0.64 meters.
    • The next rebound height will be: \[ 0.4 \times 0.64 = 0.256 \text{ meters} \]
    • It falls down 0.256 meters.
    • Finally, it will rebound to: \[ 0.4 \times 0.256 = 0.1024 \text{ meters} \]
    • It will fall down 0.1024 meters.

Now we can calculate the total distance as follows. We sum all the upward and downward distances traveled until the fifth impact with the ground.

Total Distance Calculation

Now we list down all movements until the fifth strike:

  1. First Toss (Up):

    • Up: \(4 \text{ m}\)
    • Down: \(4 \text{ m}\)

  2. Second Rebound:

    • Up: \(1.6 \text{ m}\)
    • Down: \(1.6 \text{ m}\)

  3. Third Rebound:

    • Up: \(0.64 \text{ m}\)
    • Down: \(0.64 \text{ m}\)

  4. Fourth Rebound:

    • Up: \(0.256 \text{ m}\)
    • Down: \(0.256 \text{ m}\)

  5. Fifth Rebound: (Final Up)

    • Up: \(0.1024 \text{ m}\)
    • The ball will strike the ground after it goes up \(0.1024 \text{ m}\) and comes down again, completing the motion for the fifth time.

Adding Distances

Now summing all the distances: \[ \text{Total Distance} = 4 + 4 + 1.6 + 1.6 + 0.64 + 0.64 + 0.256 + 0.256 + 0.1024 + 0.1024 \]

This can be calculated as follows:

  • Initial Height: \(4 + 4 = 8\) meters.
  • First Rebound: \(1.6 + 1.6 = 3.2\) meters.
  • Second Rebound: \(0.64 + 0.64 = 1.28\) meters.
  • Third Rebound: \(0.256 + 0.256 = 0.512\) meters.
  • Fourth Rebound: \(0.1024 + 0.1024 = 0.2048\) meters.

Now adding all these distances together: \[ \text{Total Distance} = 8 + 3.2 + 1.28 + 0.512 + 0.2048 \] Calculating this step-by-step:

  1. \(8 + 3.2 = 11.2\)
  2. \(11.2 + 1.28 = 12.48\)
  3. \(12.48 + 0.512 = 12.992\)
  4. \(12.992 + 0.2048 = 13.1968\)

Thus, the total distance traveled by the ball when it strikes the ground for the fifth time is: \[ \boxed{13.1968 \text{ meters}} \]