Prove that the bodies do not meet

Key concept: To prove two moving bodies do not meet, check whether their separation can ever be zero. Here the distance formula makes that test straightforward.

Step by step

To meet, the distance between the bodies must be $0$ for some real value of $t$.

From part (b), the distance is

$\sqrt{5t^2+40t+100}$

So we would need

$5t^2+40t+100=0$

Divide by 5:

$t^2+8t+20=0$

Now check the discriminant:

$\Delta=8^2-4(1)(20)=64-80=-16$

Since the discriminant is negative, this quadratic has no real roots. Therefore there is no real time $t$ for which the distance is zero.

Conclusion

The bodies do not meet.

$\boxed{\text{The bodies do not meet}}$

Pitfall alert

A common mistake is to assume that because the distance formula is available, the bodies must meet at the time when the expression is smallest. The minimum distance is not the same as zero distance.

Try different conditions

Another way is to complete the square:

$5t^2+40t+100=5(t^2+8t+20)=5\bigl((t+4)^2+4\bigr)$

This is always at least $20$, so the distance is always at least $\sqrt{20}$, never zero.

Further reading

discriminant, quadratic equation, minimum distance