Question

A ladybug could sting a beetle if the beetle is at most 20 cm away from the bug. One ladybug was 120 cm east of a clint stone, and moving west at 2cm/second, while a beetle 84 cm north of the same stone and moving south at 1 cm/sec.
a) Draw a chart showing the position of the two bugs at time (t hours).
b) When are they closest?
c) Will the beetle get stung?

Answer 1

at time t seconds, the distance z between the bugs is
z^2 = (120-2t)^2 + (84-t)^2
z dz/dt = -2(120-2t) - (84-t) = 5t-324
so dz/dt = 0 at t=64.8
so find z(64.8) and see whether it is greater than 20 cm

Answer 2

Did you make a diagram?
Did you have the clint stone as the origin O?

After t seconds let the southbound beetle be at P
let the ladybug be at R.
You have a right-angled triangle POR
RO = 84 - t cm
PO = 120 - 2t cm

PR^2 = (84-t)^2 + (120 - 2t)^2
2 PR d(PR)/dt = 2(84-t)(-1) + 2(120-2t)(-2)
= 0 for a max/min of PR

2(84-t)(-1) = 2(120-2t)(2)
(84-t)(-1) = (120-2t)(2)
-84 + t = 240 - 4t
5t = 324
t = 64.8 seconds

Plug that into PR^2 = (84-t)^2 + (120 - 2t)^2
and find PR
Is PR ≤ 20 ??