Derivatives and application Pythagorean: A North-South highway intersects....

Question : A North-South highway intersects an East-West highway at point P. An

automobile crosses P at 10:00 am, traveling east at a constant speed of 20 km/h.

A car north of P, twenty minutes later passes that same point, traveling south at 60

km/h. Find the approximate the minimum distance between the automobiles.

**Answer: 6.3 km**

I keep getting 7.8 km as my answer- don't know if I'm setting up the equations wrong or if the answer is wrong. Please help~ and thanks in advance :)

Re: Derivatives and application Pythagorean: A North-South highway intersects....

Hello, Errisa!

The problem has a strange answer . . .

Quote:

A north-south highway intersects an east-west highway at point P.

A car crosses P at 10:00 am, traveling east at a constant speed of 20 km/hr.

A car north of P, twenty minutes later passes that same point, traveling south at 60 km/hr.

Find the approximate the minimum distance between the automobiles.

**Answer: 6.3 km**

Code:

` |`

| 20/3 Q 20t R

P o - - - o - - - - o

| *

| *

60t | * x

| *

| *

S o

At 10:00, car drives east from at 20 km/hr.

By 10:20, it has moved to point km.

In the next hours, it has moved km to point

In the same hours, car moves from to at 60 km/hr.

. .

Let = distance between the cars,

Pythagorus: .

To minimize , set and solve.

We have: .

Negative time?

. . . The minimum distance occured two minutes *before* 10:20.

At 10:18, car was 6 miles east of point

At 10:18, car was 2 miles *north* of point

Their distance was: .

1 Attachment(s)

Re: Derivatives and application Pythagorean: A North-South highway intersects....

We need to be careful with our units. I'm going to use kilometers for distance and hours for time, with representing 10:00 am. The horizontal position of the first car relative to the intersection at time is and the vertical position of the second car is (because 20 minutes = 1/3 hours). Draw a picture:

http://mathhelpforum.com/attachment....1&d=1340072325

Using the Pythagorean Theorem, the distance between A and B turns out to be

To make the differentiation easier, let's minimize the square of the distance.

So the cars are closest at 10:18 am. At this time the distance between the cars is

Edit: Beaten by two minutes! Dang it Soroban, nice work.

Re: Derivatives and application Pythagorean: A North-South highway intersects....

:o thank you! Both of you~ I see which step I messed up on I didn't add 20/3 to 20t!

Sorry about late reply I was finishing up my other questions~

Thanks once again :D