A baseball is hit at a point 2 feet above the ground at a velocity of 100 feet per second and at an angle of 45 degrees with respect to the ground. The path of the baseball is given by the function: f(x)= -0.0032x^2+x+2.

where f(x) is the height of the baseball in feet and x is the horizontal distance from home plate in feet. What is the maximum height reached by the baseball?

recall that the vertex of a parabola is at x = -b/2a, which in this case is

x = 1/.0064 = 156.25

so, just find f(156.25)

To find the maximum height reached by the baseball, we need to determine the vertex of the parabolic function.

The equation of the parabola is given by f(x) = -0.0032x^2 + x + 2.

The vertex of a parabolic function in the form f(x) = ax^2 + bx + c is given by the formula:

x = -b / (2a).

For our function, a = -0.0032 and b = 1, so we can calculate the x-coordinate of the vertex as:

x = -1 / (2 * -0.0032) = 156.25.

To find the maximum height, we substitute the value of x back into the equation:

f(156.25) = -0.0032(156.25)^2 + (156.25) + 2.

Calculating this expression, we get:

f(156.25) = -0.0032(24414.06) + 156.25 + 2,
= -78.12 + 156.25 + 2,
= 80.13.

Therefore, the maximum height reached by the baseball is approximately 80.13 feet.

To find the maximum height reached by the baseball, we need to determine the vertex of the parabolic function represented by the given equation.

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.

Substituting these values into the formula, we get:
x = -1 / (2 * -0.0032)
x = 156.25

So, the horizontal distance from home plate when the ball reaches its maximum height is 156.25 feet.

To find the corresponding height, we substitute this value of x back into the function f(x):
f(156.25) = -0.0032 * (156.25)^2 + 156.25 + 2

Simplifying this equation, we can calculate the maximum height reached by the baseball.