BASEBALL The equation h = –0.005x2 + x + 3 describes the path of a baseball hit into the outfield, where h is the height and x is the horizontal distance the ball travels.
a. What is the equation of the axis of symmetry?
b. What is the maximum height reached by the baseball?
c. An outfielder catches the ball three feet above the ground. How far has the ball traveled horizontally when the outfielder catches it?
A. x=100
B. 53
C. 200
the axis is x = -b/2a
max height reach when x = -b/2a
solve for x when h=3. There will be two answers: one when going up, and one when going back down.
To find the answers to the given questions, we need to understand the equation and its properties. The equation h = –0.005x^2 + x + 3 is a quadratic equation that represents the path of a baseball hit into the outfield.
a. The equation of the axis of symmetry for a quadratic equation in the form ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = -0.005 and b = 1. Plugging these values into the formula, we get: x = -1 / (2 * -0.005).
To find the equation of the axis of symmetry, we substitute this value of x into the original equation: h = –0.005(-1 / (2 * -0.005))^2 + (-1 / (2 * -0.005)) + 3.
Simplifying the equation gives us: h = 3.0125.
Therefore, the equation of the axis of symmetry is x = 0.5.
b. To find the maximum height reached by the baseball, we need to look at the vertex of the quadratic equation. The vertex is the highest or lowest point of the parabolic curve.
The x-coordinate of the vertex is given by the equation x = -b / (2a), which we already calculated as x = 0.5.
To find the corresponding y-coordinate (maximum height), we substitute this value of x back into the original equation: h = –0.005(0.5)^2 + 0.5 + 3.
Simplifying the equation gives us: h = 3.0125.
Therefore, the maximum height reached by the baseball is 3.0125 units.
c. To find the horizontal distance traveled by the ball when it is caught by the outfielder, we need to find the value of x when h (height) is three feet.
We can substitute h = 3 into the equation and solve for x: 3 = –0.005x^2 + x + 3.
Rearranging the equation gives us: 0 = –0.005x^2 + x.
Factoring out x, we get: x(–0.005x + 1) = 0.
Solving for each factor separately gives us two possible solutions: x = 0 or x = 200.
Since the ball cannot have traveled a horizontal distance of x = 0 at a height of 3 feet, the correct solution is x = 200.
Therefore, the ball has traveled 200 feet horizontally when the outfielder catches it.