Understanding the algebra problem

Question

A ball is thrown upward and outward from a height of 6 feet. The height of the ball, 𝑓(𝑥), in feet, can be modeled by 𝑓(𝑥) = −0.6𝑥^2 + 2.7𝑥 + 6 where 𝑥 is the ball’s horizontal distance, in feet, from where it was thrown. What is this problem asking you to do?

Answer 1

f(x)=-.6x^2+2.7x+6

f'(x) =-1.2x +2.7=0

x = 27/12 =9/4 = 2.25 feet = horizontal distance when the ball is at maximum height

f(2.25) = -.6(2.25^2) +2.7(2.25) +6= 9.0375 feet high= max height

it hits the ground when x=6.131 feet

-.6x^2 +2.27x+6 =0

use the quadratic formula to solve for x, ignore the negative solution

those are the two main questions that come up with problems like this

find max height and time when it hits the ground

solve with calculus or algebraically find the vertex of the downward opening parabola

vertex is (2.25, 9.0375)

or graph the parabola with a graphing calculator, then find the max point =vertex

and the positive x intercept

another problem sometimes asked is what is the velocity when it hits the ground?

velocity = v = f'(x) = -1.2x +2.7

v(6.131) =-1.2(6.131)+2.7 = -4,6572 feet down per foot horizontally. but that's not the usual velocity of feet per time, so unlikely they're asking that

generally these problems are with f(t) with t= time

another question they may be asking is what is the angle the ball was thrown.It's upward and horizontally, but given the function you could calculate the angle in degrees or radians