Question

Where x is the horizontal distance in feet from the point at which the ball is thrown.How far from the child does the ball strike the ground?

Answer 1

To calculate the maximum distance that the ball takes, we can use the first and second derivatives of y to find out this. We have:

`\begin{gathered} y=-(1)/(14)x^2+4x+3 \\ \Rightarrow y^(\prime)=-(2)/(14)x+4=-(1)/(7)x+4 \\ \Rightarrow y^(\doubleprime)=-(1)/(7) \end{gathered}`

Using the second derivative criterion, we have that y'' < 0, therefore, we have a maximum in the root of the first derivative. For that, we get the following:

`\begin{gathered} y^(\prime)=0 \\ \Rightarrow-(1)/(7)x+4=0 \\ \Rightarrow(1)/(7)x=4 \\ \Rightarrow x=7\cdot4=28 \\ x=28 \end{gathered}`

Therefore, at x=28 is where the maximum distance is. Now we only substitute x=28 in y to find out:

`\begin{gathered} y=-(1)/(14)(28)^2+4(28)+3 \\ \Rightarrow y=-(1)/(14)(784)+112+3 \\ \Rightarrow y=-56+112+3=59 \\ y=59 \end{gathered}`

Finally, we have that the ball will go 59 feet from the child