maximum height and horizontal distance

Question

a football player punts a football at an inclination of 45 degrees to the horizontal at an initial velocity of 60 feet per second. the height of the football is given by the quadratic function h(x)=-32x^2/(60)^2+x+4 where x is the horizontal distance of the football from the point at which it was kicked. what was the horizontal ditance of the football from the point at which it was kicked when the height of the ball is at a maximum? what is the mazimum height of the football

Answer 1

This involves a bit of physics and some trigonometry.

 

h(x)=-32x^2/(60)^2+x+4  is the formula for vertical position as function of horizontal position.

 

We would like to identify when this quadratic function is at a maximum.  This can be done quickly using calculus, if we recognize that the maximal height occurs at a critical point of the function...namely we seek a point where the rate of change of the height is 0 (since the leading coefficient of the quadratic is negative, we know it is concave down and a critical point will be a local maximum).

 

h'(x) = -64x/60^2 + 1 = 0, leads us to x=60^2/64, as the position horizontally when the ball is at a maximum height.

 

The maximal height is then given by h(60^2/64).

 

Hope that helps...to solve this problem without the derivative and calculus, we can algebraically complete the square for h(x) to write it in the form (x+a)^2+b, for some 'a' and 'b'...and then recognize the at x=-a we have a maximum.