We are studying Quadratic formula:
A football player punts a ball. The path of the ball can be modeled by the equation y=-0.004x(squared) + x + 2.5, where x is the horizontal distance, in feet, the ball travels and y is the height, in feet, of the ball. How far from the football player will the ball land? Round to the nearest tenth of a foot.
y = -0.004x^2 + x + 2.5
the ball lands where y=0
-0.004x^2 + x + 2.5 = 0
x = 25(5+√26) = 252.5
To find the distance at which the ball will land, we need to determine the x-coordinate of the vertex of the quadratic equation representing the path of the ball. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients in the quadratic equation.
In this case, our equation is y = -0.004x^2 + x + 2.5, so the coefficient of x^2 is -0.004 and the coefficient of x is 1. Plugging these values into the formula, we have:
x = -1 / (2 * -0.004)
Simplifying the expression:
x = -1 / (-0.008)
x ≈ 125
Therefore, the ball will land at approximately x = 125 feet from the football player.
To find out how far from the football player the ball will land, we need to find the x-coordinate at which the height (y) of the ball is zero. This is because at that point, the ball will be on the ground.
To find the x-coordinate at which y=0, we need to solve the quadratic equation -0.004x^2 + x + 2.5 = 0 for x.
The quadratic equation is in the form ax^2 + bx + c = 0. In this case, a = -0.004, b = 1, and c = 2.5.
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c, we get:
x = (-(1) ± √((1)^2 - 4(-0.004)(2.5))) / (2(-0.004))
Simplifying further:
x = (-1 ± √(1 - (-0.04)(2.5))) / (-0.008)
x = (-1 ± √(1 + 0.1)) / (-0.008)
x = (-1 ± √1.1) / (-0.008)
Now, we can approximate the value of x by taking the positive square root:
x = (-1 + √1.1) / (-0.008)
Using a calculator, we find:
x ≈ 140.87
Therefore, the ball will land approximately 140.9 feet away from the football player.