The function is the following:
[tex]h(d)=-0.1d^2+1.1d+0.5[/tex].
This is a function of the height of the ball, in terms of d, the horizontal distance.
When the ball lands, h is equal to 0, so we need to find the value of d for which h(d) is 0, so we need to solve the equation:
[tex]-0.1d^2+1.1d+0.5=0[/tex].
Thus, we have a quadratic equation to solve: we use the discriminant formula!
a=-0.1, b=1.1, c=0.5, thus the discriminant is [tex]D=b^2-4ac=(1.1)^2-4(-0.1)(0.5)=1.21+0.2=1.41[/tex].
The square root of 1.41 is approximately 1.19,
thus, the roots of the equation are:
[tex]d_1= \frac{-1.1+1.19}{2(-0.1)}= \frac{0.09}{-0.2}= -0.45[/tex]
[tex]d_2= \frac{-1.1-1.19}{2(-0.1)}= \frac{-2.29}{-0.2}= 11.45[/tex]
stance must be positive, so we only consider the second answer. Thus, d=11.5 m
Answer: 11.5
