the question is in the description it wouldn't fit

Question

A ball is thrown across a playing field from a height of h = 4 ft above the ground at an angle of 45° to the horizontal at the speed of 20 ft/s. It can be deduced from physical principles that the path of the ball is modeled by the function
y = −
32
(20)2
x2 + x + 4
where x is the distance in feet that the ball has traveled horizontally.

 

 

 

Find the:

1Find the maximum height attained by the ball. (Round your answer to three decimal places.)

2Find the horizontal distance the ball has traveled when it hits the ground. (Round your answer to one decimal place.)

Answer 1

Your function is a bit garbled, but I assume it's supposed to be y = -32/(20)² * x² + x + 4

 

This is plainly a parabolic function - it fits the form y = ax² + bx + c.  Since a is negative, it is an inverted parabola, one whose opening points downwards.  Such a parabola has a maximum value, which in this case corresponds to the maximum height reached by the ball. 

 

The maximum or minimum value of a parabola occurs at the vertex, which itself is located at the x coordinate of -b/2a, based on our standard form.  With a b of 1 and an a of -32/400=-2/25=-.08, this corresponds to x = 1/.16 = 6.25.

 

We find the height the ball reaches at this point by plugging the x into our equation.

 

y = -.08*(6.25)² + 6.25 + 4 = -3.125 + 6.25 + 4 = 7.125 feet.

 

The horizontal distance covered corresponds to the x value where the ball reaches a height of 0.  I.e., it's one of the roots of the function, which can be found with the quadratic equation, x = [-b ± √(b²-4ac)]/2a

 

Plugging in our coefficients, we have

 

[-1 ± √(1 + 1.28)]/-.16 = [-1 ± √(2.28)]/-.16 = [-1 ± 1.51]/-.16 = 6.25 ± 9.44

 

There is one positive solution and one negative solution to this - for physical reasons, the positive one is the one we prefer, since we're looking for where the ball hits the ground ahead of the thrower.  Thus, our answer is x = 15.7 feet.