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.

1 Answer
Jan 2, 2018

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).
my comp

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.