A ball is dropped from the top of a 256-foot-high building. The ball is 192 feet above ground after 2 seconds, and it reaches ground level in 4 seconds.

Let the height above the ground after t seconds be,

\(\displaystyle{h}{\left({t}\right)}={a}{t}^{{{2}}}+{b}{t}+{c}\) \(\rightarrow\) (1)

Since the initial height is 256 foot, therefore \(\displaystyle{h}{\left({t}\right)}={256}\) when \(\displaystyle{t}={0}\).

Now put \(\displaystyle{t}={0}\) and \(\displaystyle{h}{\left({t}\right)}={256}\) in equation (1), we get

\(\displaystyle\Rightarrow{256}={a}{\left({0}\right)}^{{{2}}}+{b}{\left({0}\right)}+{c}\)

\(\displaystyle\Rightarrow{c}={256}\)

Since the ball is 192 feet above ground after 2 seconds, therefore \(\displaystyle{h}{\left({t}\right)}={192}\) when \(\displaystyle{t}={2}\).

Now put \(\displaystyle{t}={2}\) and \(\displaystyle{h}{\left({t}\right)}={256}\) in equation (1), we get

\(\displaystyle\Rightarrow{192}={a}{\left({2}\right)}^{{{2}}}+{b}{\left({2}\right)}+{c}\)

\(\displaystyle\Rightarrow{4}{a}+{2}{b}+{c}={192}\)

\(\displaystyle\Rightarrow{4}{a}+{2}{b}+{256}={192}\)

\(\displaystyle\Rightarrow{4}{a}+{2}{b}=-{64}\)

\(\displaystyle\Rightarrow{2}{a}+{b}=-{32}\) \(\displaystyle\rightarrow\) (2)

Since the ball reaches the ground level after 4 seconds, therefore \(\displaystyle{h}{\left({t}\right)}={0}\) when \(\displaystyle{t}={4}\).

Now put \(\displaystyle{t}={4}\) and \(\displaystyle{h}{\left({t}\right)}={0}\) in equation (1), we get

\(\displaystyle\Rightarrow{0}={a}{\left({4}\right)}^{{{2}}}+{b}{\left({4}\right)}+{c}\)

\(\displaystyle\Rightarrow{16}{a}+{4}{b}+{c}={0}\)

\(\displaystyle\Rightarrow{16}{a}+{4}{b}+{256}={0}\)

\(\displaystyle\Rightarrow{16}{a}+{4}{b}=-{256}\)

\(\displaystyle\Rightarrow{4}{a}+{b}=-{64}\) \(\displaystyle\rightarrow\) (3)

Now subtracting equation (2) from equation (3), we get

\(\displaystyle{\left({4}{a}+{b}\right)}–{\left({2}{a}+{b}\right)}=-{64}–{\left(-{32}\right)}\)

\(\displaystyle\Rightarrow{2}{a}=-{32}\)

\(\displaystyle\Rightarrow{a}=-{16}\)

Now put \(\displaystyle{a}=-{16}\) in equation (2), we get

\(\displaystyle\Rightarrow{2}{\left(-{16}\right)}+{b}=-{32}\)

\(\displaystyle\Rightarrow{b}={0}\)

Now put \(\displaystyle{a}=-{16}\), \(\displaystyle{b}={0}\) & \(\displaystyle{c}={256}\) in equation (1), we get

\(\displaystyle{h}{\left({1}\right)}=-{161}^{{{2}}}+{256}\)