I need help with this problem: The path traveled by a golf ball with a 9-iron can be modeled with the quadratic equation, y= -0.042xsquared + 5x, where x is the distance in yards from the point it was hit and y is the height of the golf ball in feet. Assume that the ground is level.
a. Find the maximum height reached by the ball.
b. How far from where it was hit does the ball hit the ground?
I really need to understand this, so please show the work. Any help you can give would be greatly appreciated.
Sure! To find the maximum height reached by the golf ball, we need to determine the vertex of the quadratic equation. The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the following formula:
Vertex x-coordinate = -b / 2a
In our case, the equation is y = -0.042x^2 + 5x, so a = -0.042 and b = 5. Plugging these values into the formula:
Vertex x-coordinate = -5 / (2 * (-0.042))
= 5 / 0.084
= 59.52
Therefore, the x-coordinate of the vertex is 59.52.
Now, to find the maximum height reached by the ball, we substitute this x-coordinate value back into the equation:
y = -0.042(59.52)^2 + 5(59.52)
= -0.042 * 3543.8304 + 297.6
= -148.7421808 + 297.6
= 148.8
So, the maximum height reached by the ball is 148.8 feet.
Moving on to part b, we need to determine the x-coordinate at which the ball hits the ground. To do this, we set y equal to zero in the equation and solve for x:
0 = -0.042x^2 + 5x
This equation can be factored as:
0 = x(-0.042x + 5)
So, either x = 0 or -0.042x + 5 = 0.
For x = 0, this represents the point of hitting the ball, so we disregard it since we are interested in where it lands.
To solve -0.042x + 5 = 0, we isolate x:
-0.042x = -5
x = -5 / (-0.042)
= 119.048
Therefore, the ball hits the ground approximately 119.048 yards away from where it was hit.
I hope this explanation helps you understand how to solve the problem. Let me know if you have any further questions!