a model rocket is launched from a roof into a large feild. 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.
A:25.80 Meters
B:37.00 Meters
C:17.24 Meters
D:16.91 Meters
My answer is D
I agree
thank you Steve
D: 16.91 Meters
Why did the rocket land there, you ask? Well, it realized that everywhere else was already occupied by rockets and decided to make its own little landing spot. It's all about finding your own space in this crowded world, even for rockets! So, D it is - 16.91 meters away from its starting point. Kudos to the rocket for getting that personal space!
To find out how far horizontally the rocket will land, we need to determine the value of x when y equals zero. This is because when the rocket lands, its height above the ground will be zero.
So, we set y equal to zero in the equation given and solve for x:
0 = -0.8x^2 + 12x + 25.8
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = -0.8, b = 12, and c = 25.8. Substituting these values into the formula:
x = (-12 ± √(12^2 - 4(-0.8)(25.8))) / 2(-0.8)
Simplifying further:
x = (-12 ± √(144 - (-82.56))) / (-1.6)
x = (-12 ± √(144 + 82.56)) / (-1.6)
x = (-12 ± √(226.56)) / (-1.6)
x = (-12 ± 15.05) / (-1.6)
Now, we can calculate two potential values for x:
x1 = (-12 + 15.05) / (-1.6) ≈ 16.91
x2 = (-12 - 15.05) / (-1.6) ≈ -1.16
Since we are interested in the horizontal distance from the starting point, we can disregard the negative value (-1.16). Therefore, the rocket will land approximately 16.91 meters horizontally from its starting point.
Hence, the correct answer is D: 16.91 Meters.
To find out how far horizontally the rocket will land, we need to determine the value of x when the rocket hits the ground. The rocket hits the ground when y = 0.
We can set the equation y = -0.8x^2 + 12x + 25.8 equal to 0 and solve for x.
-0.8x^2 + 12x + 25.8 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = -0.8, b = 12, and c = 25.8.
Substituting these values into the quadratic formula:
x = (-12 ± √(12^2 - 4(-0.8)(25.8))) / (2(-0.8))
Simplifying further:
x = (-12 ± √(144 + 82.56)) / (-1.6)
x = (-12 ± √226.56) / (-1.6)
x = (-12 ± 15.05) / (-1.6)
We have two possible solutions:
1. x = (-12 + 15.05) / (-1.6) = 3.05 / (-1.6) = -1.90625
2. x = (-12 - 15.05) / (-1.6) = -27.05 / (-1.6) = 16.90625
Since distance cannot be negative in this context, we discard the first solution (-1.90625) and choose the second solution (16.90625).
Rounding this value to the nearest hundredth, we get approximately 16.91 meters.
Therefore, the correct answer is D: 16.91 meters.