Question

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 23 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 7 PM?
The distance is changing at _____________ knots.

(Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

hey @pitamar do you how to solve this problem?

Answer 2

Ok, let's try =)
Seems like what we need here is phytagorean theorem and derivatives of polynomials. are you familiar with them both?

Answer 3

i think so. pythagorean theorem is x^2 +y^2 = s^2 and to find the derivatives you do something like 2sds/dt = 2xdx/dt + 2ydy/dt. is that right?

Answer 4

sorry was cooking heh
well, let's start with the pythagorean theorem first. We have:

right?

Answer 5

ok yes

Answer 6

Ok, so we can say using pythagorean theorem that:
$$ (\text{distance})^2 = (20 + 23t)^2 + (21t)^2 $$
Right?

Answer 7

yes

Answer 8

wanna multiply it and see what you get? =)

Answer 9

d^2 = 400 + 920t +529t^2

Answer 10

ehmm, why 529? We have \((23t)^2 + (21t^2) = t^2 \cdot ( (23)^2 + (21)^2) = 970t^2\)
The whole thing is pretty much
$$ (\text{distance})^2 = 970t^2 + 920t + 400 $$

Answer 11

So, we can take root and have
$$
\text{distance} = \sqrt{970t^2 + 920t + 400}
$$Because distance can only be positive
ok?

Answer 12

ohhh i forgot to add the 21t^2. i just foiled the first part

Answer 13

lol that's ok =)
So now let's call distance 'D' and we want to find the rate of change of the distance 'D' with respect to time 't'. Or dD/dt. Which is the derivative of what we have here with respect to t
$$
\frac{dD}{dt} = \left( \sqrt{970t^2 + 920t + 400} \right)' = ?
$$

Answer 14

1/2 (1/(sqrt(970t^2 + 920t + 400)) * 1940t+920

Answer 15

ye seems ok. now we want to find that in 7PM. we said t0 is at noon, so 7pm is 7 hours later where t=7

Answer 16

just plug in? 1/2 (1/(sqrt(970(7)^2 + 920(7) + 400)) * 1940(7)+920

Answer 17

yep

Answer 18

i got 31.09269309

Answer 19

ye me too

Answer 20

ok yay thank you!!

Answer 21

sure, hope it's right =)

Answer 22

it is :)