The parabola, B, represents a shuriken's maximum throwing distance, which was 60 meters of the enemy ninja, A, from a height of 7 meters. Figure out the trait of the enemy ninja that can be observed from the ellipse formed by the shuriken, and figure out the maximum throwing distance on flat ground. Explain your reasoning.
To figure out the trait of the enemy ninja that can be observed from the ellipse formed by the shuriken, we need to understand the nature of the parabola and its relationship with the ellipse.
A parabola is a U-shaped curve that is symmetric around a single axis. It has a vertex, which is the highest or lowest point on the curve, and it opens either upwards or downwards. In this case, the parabola represents the maximum throwing distance of the shuriken, so it opens upwards.
An ellipse, on the other hand, is a closed curve that is symmetric around two axes. It has two foci points, and the sum of the distances from any point on the ellipse to the two foci points is constant.
In this scenario, the shuriken's maximum throwing distance forms an ellipse with the enemy ninja's position at one of its foci points. The other foci point would be located at the same distance from the enemy ninja as the maximum throwing distance on flat ground. To figure out this maximum throwing distance on flat ground, we need to analyze the properties of the parabola.
From the given information, we know that the shuriken's maximum throwing distance is 60 meters from the enemy ninja. We also know that the height from which it was thrown is 7 meters.
To find the equation of the parabola, we can use the standard form: y = ax^2 + bx + c. In this case, since the vertex is the maximum point, we have a > 0. Let's define the parabola's vertex as (h, k).
The vertex form of the parabola equation is: y = a(x - h)^2 + k.
Since the vertex (h, k) represents the maximum throwing distance of the shuriken, we can substitute the given values to get:
y = a(x - h)^2 + k,
y = a(x - 0)^2 + 7,
y = ax^2 + 7.
Now we have the equation y = ax^2 + 7.
Since the maximum throwing distance is 60 meters, we can substitute the corresponding coordinates (x, y) = (30, 60) into the equation:
60 = a(30)^2 + 7,
60 = 900a + 7,
900a = 53,
a = 53/900.
Now, we have the value of a, and the equation of the parabola becomes y = (53/900)x^2 + 7.
To find the maximum throwing distance on flat ground, we need to determine the x-intercept of the parabola, which represents the point where the shuriken hits the ground. At this point, y = 0.
Setting y = 0 in the equation (53/900)x^2 + 7 = 0, we can solve for x:
(53/900)x^2 + 7 = 0,
(53/900)x^2 = -7,
x^2 = -(7/(53/900)),
x^2 = -7 * (900/53),
x^2 ≈ -118.68.
The x^2 value is negative, which means there are no real solutions for x and, therefore, no real x-intercepts. This implies that the parabola does not intersect the x-axis, and the shuriken does not hit the ground on flat ground.
Thus, the trait of the enemy ninja that can be observed from the ellipse formed by the shuriken is that the shuriken can reach a maximum throwing distance of 60 meters from the enemy ninja, but it will not hit the ground on flat ground.