Three ballet dancers are positioned on stage. Leo is 1 meter straight behind Eve and 2 meters
directly left of Abby. When the music begins, Leo twirls to Abby's position, then leaps to Eve's
position, and finally walks back to his original position. How far did Leo travel? If necessary,
round to the nearest tenth.
To solve the problem, we need to find the distances Leo traveled during his twirl, leap, and walk.
First, let's find the distance between Leo and Abby before the twirl. Using the Pythagorean theorem, we have:
distance between Leo and Abby = √(1^2 + 2^2) = √5
Next, let's find the distance between Abby and Eve. Since we know that Leo is straight behind Eve, we can use the Pythagorean theorem again to find the distance between Abby and Eve:
distance between Abby and Eve = √(2^2 + 1^2) = √5
Now we can find the distance Leo traveled during his twirl. He moved from his original position to Abby's position, which we know is √5 meters away.
During his leap, Leo moved from Abby's position to Eve's position. Since we know the distance between Abby and Eve is also √5, we can use the Pythagorean theorem one more time to find the distance Leo traveled during his leap:
distance traveled during leap = √[(√5)^2 + (√5)^2] = √20 = 2√5
Finally, during his walk, Leo traveled 1 meter back to his original position.
Adding up the three distances, we get:
total distance traveled = √5 + 2√5 + 1 ≈ 4.2 meters
So Leo traveled approximately 4.2 meters during his twirl, leap, and walk.
To find the distance Leo traveled, we need to calculate the distance between his initial position and each of the three positions he moved to.
1. Distance between Leo and Abby:
Leo is positioned 2 meters directly left of Abby. Since Abby is stationary, Leo will need to move 2 meters to reach Abby's position.
2. Distance between Leo and Eve:
Leo initially starts 1 meter straight behind Eve. After twirling to Abby's position, he will have to travel in a circular path to reach Eve's position. Let's assume the distance Leo traveled while twirling is x meters. After twirling, Leo will then have to walk 1 meter forward to reach Eve's position.
3. Distance between Leo and his initial position:
After reaching Eve's position, Leo needs to walk back to his original position. Since he walked 1 meter forward to reach Eve's position, he will need to walk 1 meter backward to return to his initial position.
Now, let's calculate the distance traveled by Leo step-by-step:
1. Distance between Leo and Abby: 2 meters
2. Distance while twirling from Abby to Eve: x meters
3. Distance between Leo and Eve: x + 1 meter
4. Distance from Eve back to Leo's initial position: 1 meter
Total distance traveled by Leo = Distance to Abby + Distance while twirling + Distance to Eve + Distance back to initial position
Total distance traveled by Leo = 2 + (x + 1) + 1
Since Leo twirled from one position to another, the total distance traveled during the twirl is equal to the circumference of the circular path he followed. The circumference can be calculated using the formula: circumference = 2πr.
In this case, the radius of the circular path is the distance between Abby and Eve, which is 1 meter. So, the distance traveled during the twirl is 2π(1) = 2π meters.
Now, let's substitute the value of x in the equation and calculate the total distance traveled by Leo:
Total distance traveled by Leo = 2 + (2π + 1) + 1
Total distance traveled by Leo = 4 + 2π
To find an approximate value, let's use 3.14 as an approximation for π:
Total distance traveled by Leo ≈ 4 + 2(3.14)
Total distance traveled by Leo ≈ 4 + 6.28
Total distance traveled by Leo ≈ 10.28
Therefore, Leo traveled approximately 10.3 meters.