the path of a ball is given by y=-.05+3x^2+3x+5, where y is the height (in feet) of the ball and x is the horizontal distance (in feet) from where the ball was thrown. Find the maximum height of the ball.

To find the maximum height of the ball, we need to determine the vertex of the function. The vertex represents the maximum point on the graph.

The given equation represents a quadratic function in the form y = ax^2 + bx + c, where a, b, and c are constants.

Comparing the equation with the standard form, we have:
a = 3
b = 3
c = -0.05 + 5 = 4.95

The x-coordinate of the vertex can be found using the formula x = -b / (2a). In this case, x = -3 / (2*3) = -1/2.

Substituting the x-coordinate into the equation will give us the y-coordinate of the vertex: y = -0.05 + 3(-1/2)^2 + 3(-1/2) + 4.95.

Evaluating the expression, we get:
y = -0.05 + 3(1/4) - 3/2 + 4.95
y = -0.05 + 0.75 - 1.5 + 4.95
y = 5.15

Therefore, the maximum height of the ball is 5.15 feet.

To find the maximum height of the ball, we need to determine the vertex of the quadratic equation.

The quadratic equation is in the form of y = ax^2 + bx + c, where a = 3, b = 3, and c = 5.

The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula:

x = -b / (2a)

Let's substitute the values into the formula:

x = -3 / (2 * 3)
x = -1/2

To find the maximum height (y), substitute the value of x into the equation:

y = -0.05 + 3(-1/2)^2 + 3(-1/2) + 5

Simplifying the equation:

y = -0.05 + 0.75 - 1.5 + 5
y = 4.2

Therefore, the maximum height of the ball is 4.2 feet.