Question
A model rocket is launched from a roof into a large field.The path of the rocket can be modeled by the solution.Y equals negative zero point 06 x squared plus nine point six x
Answers
Answered by
GPT 3.5
minus three point two, where Y represents the height of the rocket above the ground in meters and x represents the horizontal distance from the starting point in meters.
This equation represents a quadratic function, and it can be used to calculate the height of the rocket at any given horizontal distance from the starting point. By analyzing the function, we can determine the maximum height reached by the rocket and the horizontal distance it travels before hitting the ground.
To find the maximum height of the rocket, we need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form ax^2 + bx + c is given by the formula x = -b/2a.
In this case, a = -0.06 and b = 9.6. Therefore, the x-coordinate of the vertex is x = -9.6 / (2*(-0.06)) = 80. The maximum height of the rocket is then given by plugging x = 80 into the function:
Y = -0.06(80)^2 + 9.6(80) - 3.2
Y = -384 + 768 - 3.2
Y = 380.8 meters
Therefore, the maximum height reached by the rocket is 380.8 meters.
To determine the horizontal distance the rocket travels before hitting the ground, we need to find the x-intercepts of the function. The x-intercepts correspond to the points where the rocket hits the ground.
To find the x-intercepts, we set Y = 0 and solve for x:
0 = -0.06x^2 + 9.6x - 3.2
0 = 0.06x^2 - 9.6x + 3.2
This equation can be solved using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in a = 0.06, b = -9.6, and c = 3.2, we get x ≈ 15.4 and x ≈ 34.6.
Therefore, the rocket travels approximately 15.4 meters and 34.6 meters before hitting the ground.
This equation represents a quadratic function, and it can be used to calculate the height of the rocket at any given horizontal distance from the starting point. By analyzing the function, we can determine the maximum height reached by the rocket and the horizontal distance it travels before hitting the ground.
To find the maximum height of the rocket, we need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form ax^2 + bx + c is given by the formula x = -b/2a.
In this case, a = -0.06 and b = 9.6. Therefore, the x-coordinate of the vertex is x = -9.6 / (2*(-0.06)) = 80. The maximum height of the rocket is then given by plugging x = 80 into the function:
Y = -0.06(80)^2 + 9.6(80) - 3.2
Y = -384 + 768 - 3.2
Y = 380.8 meters
Therefore, the maximum height reached by the rocket is 380.8 meters.
To determine the horizontal distance the rocket travels before hitting the ground, we need to find the x-intercepts of the function. The x-intercepts correspond to the points where the rocket hits the ground.
To find the x-intercepts, we set Y = 0 and solve for x:
0 = -0.06x^2 + 9.6x - 3.2
0 = 0.06x^2 - 9.6x + 3.2
This equation can be solved using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in a = 0.06, b = -9.6, and c = 3.2, we get x ≈ 15.4 and x ≈ 34.6.
Therefore, the rocket travels approximately 15.4 meters and 34.6 meters before hitting the ground.