The path of a ball is given by y=-1/20x^2+3x+5, where y is the height in feet and x is the horizontal distance in feet. find the maximum height of the ball and the horizontal distance at that height

The path of the ball is given by where y is the height in feet and x is the horizontal distance in feet.

We will use the second derivative test which states,

If a function has a critical point for which f′(x) = 0 and the second derivative is positive at this point, then f has a local minimum. If, the function has a critical point for which f′(x) = 0 and the second derivative is negative at this point, then f has local maximum.

Firstly, we will calculate the first derivative of the given function,

To find the horizontal distance, let y'=0

Now let us consider the second derivative of the given function,

Since the second derivative that is y'' is negative.

Therefore, x=30 is the maximum horizontal distance.

Substituting x =30 in the given function to get the maximum height,

y=50 is the maximum height in feet.

the 3rd option is right

step-by-step explanation: