The cluster's height above the valley floor can be modeled by the equation:
[tex]h(t)=-4 t^{2} +64t+320[/tex]
When the cluster seeds will reach the valley floor, the height of the cluster seeds above the valley floor will be 0. So substituting h(t) = 0 in the above equation we can find the time when the cluster seeds will reach the valley floor.
[tex]h(t)=0=-4t^{2} +64t+320 \\ \\
-4t^{2} +64t+320=0 \\ \\
-4(t^{2}-16t-80)=0 \\ \\
t^{2}-16t-80=0 \\ \\
t^{2} -20t+4t-80=0 \\ \\
t(t-20)+4(t-20)=0 \\ \\
(t+4)(t-20)=0 \\ \\
t=-4,t=20[/tex]
Since t represents the amount of time, it cannot have a negative value. So the only acceptable value of t is t=20
Thus it will take 20 seconds for the cluster of seeds to reach the valley floor. Thus the correct answer is option C
