Fred hits a baseball 3 ft above ground at a velocity of 100 ft per sec and at an Angle of 45 degrees with respect to level ground. The path of the baseball is given by the function f(x)=-0.0032x^2+x+3 where f(x) is the height of the baseball in ft and x is the distance from home plate in ft. What is the maximum height reached by the baseball?
Well, I must say, Fred really knows how to send a baseball flying! To find the maximum height reached by the baseball, we need to determine the vertex of the parabola described by the function f(x).
The vertex of a parabola in the form f(x) = ax^2 + bx + c is given by the equation x = -b/2a. In this case, a = -0.0032 and b = 1, so we can calculate the x-coordinate of the vertex as follows:
x = -b/2a = -(1) / (2 * -0.0032) = 156.25 ft
Now that we have the x-coordinate of the vertex, we can substitute it back into the function f(x) to find the corresponding y-coordinate:
f(156.25) = -0.0032(156.25)^2 + 156.25 + 3 ≈ 200.78 ft
So, according to my calculations, the maximum height reached by the baseball is approximately 200.78 feet. It's almost as if the baseball wanted to give the sky a high-five!
To find the maximum height reached by the baseball, we need to determine the vertex of the parabola given by the function f(x).
The vertex of a parabola in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).
In this case, a = -0.0032 and b = 1.
x = - (1) / (2 * (-0.0032)) = 156.25 ft
Now we substitute this value of x back into the function f(x) to find the maximum height:
f(156.25) = -0.0032(156.25)^2 + 156.25 + 3
= -0.0032(24375.625) + 156.25 + 3
= -78.0002 + 156.25 + 3
= 81.2498 ft
Therefore, the maximum height reached by the baseball is approximately 81.2498 ft.