Answer: 22.27 feet or 31.68 feet, depending on which direction they are throwing the ball.
Explanation:
Let's call the distance between Simon and Kaylee "x". We can use the Pythagorean theorem to set up two equations based on the given distances:
x^2 + 30^2 = d1^2 (where d1 is the distance between Simon and Whitney)
x^2 + 34^2 = d2^2 (where d2 is the distance between Kaylee and Whitney)
We can solve for d1 and d2 by taking the square root of both sides of each equation:
d1 = sqrt(x^2 + 900)
d2 = sqrt(x^2 + 1156)
Since we know that d1 + d2 = 30 + 34 = 64 feet (the total distance between Simon and Kaylee), we can substitute the expressions for d1 and d2 into this equation:
sqrt(x^2 + 900) + sqrt(x^2 + 1156) = 64
We can square both sides of the equation to eliminate the square roots:
x^2 + 900 + 2sqrt((x^2 + 900)(x^2 + 1156)) + x^2 + 1156 = 4096
Simplifying:
2x^2 + 2056 + 2sqrt((x^2 + 900)(x^2 + 1156)) = 4096
Subtracting 2056 from both sides:
2x^2 + 2sqrt((x^2 + 900)(x^2 + 1156)) = 2040
Dividing both sides by 2:
x^2 + sqrt((x^2 + 900)(x^2 + 1156)) = 1020
Squaring both sides again to eliminate the square root:
x^4 + 2052x^2 + 656100 = 1040400
Simplifying:
x^4 + 2052x^2 - 384300 = 0
This is a quadratic equation in x^2. We can solve for x^2 using the quadratic formula:
x^2 = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 2052, and c = -384300. Plugging in these values:
x^2 = (-2052 ± sqrt(2052^2 - 4(1)(-384300))) / 2(1)
x^2 = (-2052 ± sqrt(9248256)) / 2
x^2 = (-2052 ± 3044) / 2
x^2 = 496 or x^2 = 1004
Taking the positive square root of both sides (since x represents a distance, it must be positive), we get:
x = sqrt(496) or x = sqrt(1004)
x ≈ 22.27 feet or x ≈ 31.68 feet
So the distance between Simon and Kaylee is approximately 22.27 feet or 31.68 feet, depending on which direction they are throwing the ball.