The function is the following:
![h(d)=-0.1d^2+1.1d+0.5](https://tex.z-dn.net/?f=h%28d%29%3D-0.1d%5E2%2B1.1d%2B0.5)
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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:
![-0.1d^2+1.1d+0.5=0](https://tex.z-dn.net/?f=-0.1d%5E2%2B1.1d%2B0.5%3D0)
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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
![D=b^2-4ac=(1.1)^2-4(-0.1)(0.5)=1.21+0.2=1.41](https://tex.z-dn.net/?f=D%3Db%5E2-4ac%3D%281.1%29%5E2-4%28-0.1%29%280.5%29%3D1.21%2B0.2%3D1.41)
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The square root of 1.41 is approximately 1.19,
thus, the roots of the equation are:
![d_1= \frac{-1.1+1.19}{2(-0.1)}= \frac{0.09}{-0.2}= -0.45](https://tex.z-dn.net/?f=d_1%3D%20%5Cfrac%7B-1.1%2B1.19%7D%7B2%28-0.1%29%7D%3D%20%5Cfrac%7B0.09%7D%7B-0.2%7D%3D%20-0.45)
![d_2= \frac{-1.1-1.19}{2(-0.1)}= \frac{-2.29}{-0.2}= 11.45](https://tex.z-dn.net/?f=d_2%3D%20%5Cfrac%7B-1.1-1.19%7D%7B2%28-0.1%29%7D%3D%20%5Cfrac%7B-2.29%7D%7B-0.2%7D%3D%2011.45)
stance must be positive, so we only consider the second answer. Thus, d=11.5 m
Answer: 11.5