Problem involving intersection of a line and a circle

[Serious Max]

Homework Statement

Erik’s disabled sailboat is floating stationary 3 miles East and 2 miles North of Kingston. The sailboat has a radar scope that will detect any object within 3 miles of the sailboat. A ferry leaves Kingston heading toward Edmonds at 12 mph. Edmonds is 6 miles due east of Kingston. After 20 minutes the ferry turns heading due South. Ballard is 8 miles South and 1 mile West of Edmonds. Impose coordinates with Ballard as the origin.

When does the ferry enter the radar zone?

2. Relevant graph

The Attempt at a Solution

I wanted ask if the distance to the origin has to be an absolute value in these kind of problems?

If the equation of a circle is (x-h)^2+(y-k)^2=r^2 with h and k being the coordinates of the center of a circle, then if I x=-5, h=-2 I am going to have -5+2=-3 and get negative values (for time) with this equation ((-5+2)-12t)^2+(8-10)^2=9.

It only works if the value is positive (absolute): ((|-5+2|)-12t)^2+(8-10)^2=9, or I guess then it should look like this ((|-5+2|)-12t)^2+(|8-10|)^2=9

Also another totally silly question: where does this 12t come from? As I understand it sets the whole thing in motion but I don't quite see how it's legal, if x and h are already set in the equation. I referenced another similar problem and that's why I don't completely understand all the steps.

 

[NascentOxygen]

While traveling East, the ferry at time t hours will be located at x=-5 +12t

 

[verty]

I know the question says "impose Ballard as the origin" but that doesn't constrain how you calculate the answer, it only constrains how you give the answer. So my method for this is to place the sailboat at the origin, find the point of intersection (now easier), translate it back to the Ballard plane.

Distance is always non-negative, this is a rule for metrics which are distance functions.

 

[Serious Max]

While traveling East, the ferry at time t hours will be located at x=-5 +12t

Okay, I guess it's the answer to the last question. Thank you. Btw, is it "+" because the ferry (point) travels towards positive values? Otherwise it would have been a "-"?

Actually this equation works well (-5+12t+2)^2+(8-10)^2=9 :)

 

[Serious Max]

I know the question says "impose Ballard as the origin" but that doesn't constrain how you calculate the answer, it only constrains how you give the answer. So my method for this is to place the sailboat at the origin, find the point of intersection (now easier), translate it back to the Ballard plane.

Distance is always non-negative, this is a rule for metrics which are distance functions.

Hey yes, good idea. Didn't think about it. All these +- signs are quite confusing.

 

[NascentOxygen]

Okay, I guess it's the answer to the last question. Thank you. Btw, is it "+" because the ferry (point) travels towards positive values? Otherwise it would have been a "-"?

Yes, that's right.

BTW, I solved this using a triangle; no need for the circle equation.

Actually this equation works well (-5+12t+2)^2+(8-10)^2=9 :)

Your answer is ...?

 

[Serious Max]

0.063661 hours

 

[NascentOxygen]

0.063661 hours

Ditto.

 

[Serious Max]

What's your equation, could you reveal?

 

[SpecialPerson]

Now that we have gotten the answer, I'll show you my solution.

Let the point where the ferry enters the radar zone be (a, 8)
The distance from (a, 8) to (- 2, 10) is 3 miles.
We require sqrt[(a + 2)^2 + 2^2] = 3
(a + 2)^2 + 4 = 9
(a + 2)^2 = 5
a + 2 = ± sqrt 5
a = ± sqrt 5 - 2 = approx. 0.24 or - 4.24
a = - 4.24 corresponds to the point of entry;
a = 0.24 corresponds to the point at which it would leave if it didn't turn south.
The point where a = - 4.24 is (5 - 4.24) = 0.76 miles east of Kingston.
The ferry travels at 12 mph.
Thus the time at which the ferry enters the radar zone is (0.76/12) hours
That's 4 minutes after