A model rocket is launched from a roof into a large field. The path of the rocket can be modeled by the equation y=-0.8x^2+12x+25.8 where x is the horizontal distance, in meters, from the starting point on the roof and y is the height, in meters, of the rocket above the ground.
How far horizontally from its starting point will the rocket land? Round your answer to the nearest hundredth.
The rocket will land when it hits the ground, meaning y = 0. Therefore, we can set the equation equal to 0 and solve for x:
-0.8x^2 + 12x + 25.8 = 0
Using the quadratic formula, we get:
x = (-12 ± sqrt(12^2 - 4(-0.8)(25.8))) / (2(-0.8))
x = (-12 ± sqrt(584.64)) / (-1.6)
x ≈ -1.29 or x ≈ 15.79
Since the rocket was launched from a roof, we can disregard the negative solution. Therefore, the rocket will land approximately 15.79 meters horizontally from its starting point on the roof. Rounded to the nearest hundredth, this is 15.78 meters.
See the correct solution in the post above this
I apologize, you are correct. The correct solution is already given above.
To find the distance horizontally from the starting point where the rocket will land, we need to find the x-coordinate when the height, y, is equal to 0.
Given the equation y = -0.8x^2 + 12x + 25.8, we can set y equal to 0:
0 = -0.8x^2 + 12x + 25.8
This equation is in the form of a quadratic equation, so we can solve it by factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation 0 = -0.8x^2 + 12x + 25.8, a = -0.8, b = 12, and c = 25.8.
Plugging these values into the quadratic formula, we have:
x = (-12 ± √(12^2 - 4(-0.8)(25.8)))/(2(-0.8))
Simplifying this equation, we get:
x = (-12 ± √(144 + 82.56))/(-1.6)
x = (-12 ± √(226.56))/(-1.6)
x = (-12 ± 15.05)/(-1.6)
Now we can solve for the two possible values of x:
x₁ = (-12 + 15.05)/(-1.6) = 1.90625
x₂ = (-12 - 15.05)/(-1.6) = 16.905
Since we are looking for the horizontal distance, we are only interested in positive values of x. Therefore, the rocket will land approximately 1.91 meters horizontally from its starting point (on the roof).