Question

Does the ball hits the ceiling?

A person throws a ball from the height `h` m. This height can can be can be modelled in relation to the horizontal distance from the point it was thrown by the quadratic equation: `h=-3/10x^2+5/2x+3/2`

The hall has a sloping ceiling which can be modelled with equation:
`h=15/2-1/5x`

Show whether the model predicts that the ball will hit the ceiling.

Answer 1

The model predicts that the ball will hit the ceiling

Explanation:

It can be predicted in a number of ways. I choose part Graphical part Analytic method.
Draw a graph between distance moved `x` Vs height `h` as modeled. It looks like graph below which has maximum`"*"` at `(4.71,6.708)`.

Calculate height of ceiling at value of `x` of this maximum point by inserting in the second equation.

`h=15/2-x/5`
`h_((x=4.17))=15/2-4.17/5=6.666`

We see that height of roof is less than the height value of maximum point on the `x,h` curve.

.-.-.-.-.-.-.-.-.-

`"*"`The local maximum for quadratic equation can also be found out analytically by seting first differential of `h` with respect to `x=0`, and solving for `x`. One needs to confirm that it is actually a maximum by finding out value of second derivative at the solution point and confirming its value to be negative.

`h=-(3x^2)/10+(5x)/2+3/2`
`(dh)/dx=-6/10x+5/2`
`0=-6/10x+5/2`
`=>x=5/2xx5/3`
`=>x=25/6=4.17`

`(d^2h)/dx^2=-6/10`

It is `-ve`, hence the point is maximum.