Question 1196416
{{{y =-(32/20^2)x^2 +x+5}}}..........simplify
{{{y =-(32/400)x^2 +x+5}}} 
{{{y =-(2/25)x^2 +x+5}}}



(1) If the ball traveled horizontally 10ft, what is the height?

{{{x=10}}}
{{{y =-(2/25)10^2 +10+5}}}
{{{y =7ft}}}

If the ball traveled horizontally {{{10ft}}}, the height is {{{7ft}}}.


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


This is plainly a parabolic function - it fits the form {{{y = ax^2 + 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 {{{b =1 }}}and an {{{a = -(2/25)}}} we have

{{{-b/2a=-1/(2(-2/25))=25/4=6.25}}}

 this corresponds to {{{x = 6.25}}}

 
We find the height the ball reaches at this point by plugging the {{{x}}} into our equation.
  
{{{y = -(2/25)*(6.25)^2 + 6.25 + 5 = 8.125ft}}}.


The maximum height attained by the ball is {{{8.125ft}}}.


(3)

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

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 +- sqrt(b^2-4ac))/(2a)}}}
 
Plugging in our coefficients, we have
 

{{{x = (-1 +- sqrt(1^2-4*(-(2/25))*5))/(2(-(2/25)))}}}

{{{x = (-1 +-sqrt(1+8/5))/(-0.16)}}}

{{{x = (-1 +- sqrt(2.6))/(-0.16)}}}

{{{x=-6.25 (-1 +- 1.61245)}}}

{{{x= 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= 6.25 +9.44}}}

{{{x = 15.7ft}}}

the horizontal distance the ball has traveled when it hits the ground is {{{ 15.7ft}}}