A model rocket is launched from a roof into a large field. The path of the rocket can be modeled by the equation y = -0.06x^2 + 9.6x + 5.4 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
4.30 m
160.56 m
160.23 m
13.94 m
find the zeros of that parabola, use the + one
(I turned the parabola upside down because I like positive x^2 term)
.06 x^2 - 9.6 x - 5.4 = 0
use x = [-b +/- sqrt(b^2-4ac) ]/2a
while I use wolfram alpha
http://www.wolframalpha.com/input/?i=+.06+x^2+-+9.6+x+-+5.4+%3D+0++
to get 160.56
To find how far horizontally the rocket will land, we need to determine the value of x when y is equal to zero. This is because at the landing point, the height of the rocket will be zero.
Given the equation y = -0.06x^2 + 9.6x + 5.4, we can set y equal to zero:
0 = -0.06x^2 + 9.6x + 5.4
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.06, b = 9.6, and c = 5.4.
Plugging in these values into the quadratic formula, we get:
x = (-9.6 ± √(9.6^2 - 4(-0.06)(5.4))) / (2(-0.06))
x = (-9.6 ± √(92.16 + 1.296)) / (-0.12)
x = (-9.6 ± √93.456) / (-0.12)
Now we can calculate the two possible values of x:
x = (-9.6 + √93.456) / (-0.12) ≈ 13.94 meters
x = (-9.6 - √93.456) / (-0.12) ≈ -160.23 meters
Since the rocket can't land at a negative distance, the rocket will land approximately 13.94 meters horizontally from its starting point.
Therefore, the correct answer is 13.94 m.
To determine the horizontal distance the rocket will land from its starting point, we need to find the value of x when y equals zero. This means we need to solve the equation -0.06x^2 + 9.6x + 5.4 = 0.
To solve a quadratic equation like this, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = -0.06, b = 9.6, and c = 5.4.
Plugging in these values, we get: x = (-9.6 ± √(9.6^2 - 4*(-0.06)*5.4)) / (2*(-0.06)).
Simplifying further, we have: x = (-9.6 ± √(92.16 - (-1.296))) / (-0.12).
Continuing to simplify, we have: x = (-9.6 ± √(92.16 + 1.296)) / (-0.12).
Further simplification gives: x = (-9.6 ± √(93.456)) / (-0.12).
Taking the square root of 93.456 gives approximately 9.668.
We can then have two solutions for x: x = (-9.6 + 9.668) / (-0.12) and x = (-9.6 - 9.668) / (-0.12).
Evaluating each solution separately, we get:
x = (-0.068) / (-0.12) ≈ 0.57 and x = (-19.268) / (-0.12) ≈ 160.56.
Therefore, the rocket will land approximately 160.56 meters horizontally from its starting point.
Hence, the correct answer is 160.56 m.