The path of a cannonball is a parabola modeled by the equation y= 55 + x - 0.002x^2, where x and y are both measured in feet. In this model, the cannonball starts at the point (0,55) and travels to the right. The ground is represented by the x-axis. Determine the horizontal distance traveled by the ball before it hits the ground.
just set y=0 and solve for x. The usual quadratic equation methods.
To determine the horizontal distance traveled by the ball before it hits the ground, we need to find the x-coordinate where the cannonball intersects the x-axis (i.e., when y = 0).
The equation given to model the path of the cannonball is: y = 55 + x - 0.002x^2.
Setting y to 0 and solving for x:
0 = 55 + x - 0.002x^2
Rearranging the equation:
0.002x^2 - x - 55 = 0
Now, we can solve this quadratic equation for x. We can use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 0.002, b = -1, and c = -55. Plugging these values into the formula, we get:
x = (-(-1) ± √((-1)^2 - 4(0.002)(-55))) / (2 * 0.002)
Simplifying:
x = (1 ± √(1 + 0.44)) / 0.004
x = (1 ± √(1.44)) / 0.004
Simplifying the square root:
x = (1 ± 1.2) / 0.004
x = 2.2 / 0.004 or -0.2 / 0.004
Calculating the values:
x = 550 or -50
Since we are interested in the distance traveled to the right, we discard the negative solution. Therefore, the cannonball travels a horizontal distance of 550 feet before it hits the ground.