Math
posted by Lily .
Can someone please go over the steps to solve this problem? I know I need to find where the tugboat intersects the radar but I'm not sure exactly how to do this.
A radar buoy detects any boats within a radius of 12 miles. A tugboat starts at a location 18 miles SOUTH and 10 miles EAST of the radar buoy. The tugboat travels at a constant speed of 15 mph. The tugboat travels on a straight line toward the NORTHERNMOST point of the radar region. When the tugboat is directly EAST of the buoy, it turns and travels DUE NORTH until it exits the radar region.
How long (in hours) is the tugboat in the radar region?

Use coordinate geometry.
let the radar region be represented by the circle
x^2 + y^2 = 144
and the initial position P(10,18) , heading to (0,12)
so find the equation of that line:
slope = (12+18)/(0+10) = 3 , with a yintercept of 12
so first path : y = 3x + 12
where does it cross the circle:
x^2 + (3x+12)^2 = 144
x^2 + 9x^2 + 72x + 144 = 144
10x^2 + 72x = 0
x(10x + 72) = 0
x = 0 or x = 7.2
we know that the x=0 is our point (0,12) so
if x = 7.2
y = 3(7.2) + 12 = 9.6
So the intersection of his path with the radar region is (7.2, 9.6)
I crosses the xaxis at x = 4 , (let y=0 in y = 3x+12 )
So first leg is from (7.2 , 9.6) to (4,0) , then it heads north
so let x = 4 in the circle equation:
16 + y^2 = 144
y = √128 , which is the distance it travels northbound within the region.
Finally find the distance between (7.2, 9.6) and (4,0) , add on √128 for the total distance.
Since Time = distance/rate
divide your total distance be 15 mph to get the total time.
Good luck. 
Just realized that I read the EAST as WEST .
however, because of the symmetry of the diagram, the final answer would be the same.
P should be (10,18)
equation of first path should by y = 3x+12
crosses the circle at (7.2, 9.6)
xintercept is x = 4
first leg from (7.2, 9.6) to (4,0)
(because of the squaring in the distance formula, the results do not change)
Respond to this Question
Similar Questions

relative velocity
A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 159km away. The radar antenna is pointing upward at an angle of 59.3 from the ground.Fine the x and y components(in … 
MATH
A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 4 km and climbs at an angle of 45 degrees. At what rate, in km/min, is the distance from the plane to the radar station increasing … 
Airplane & Radar
(Airplane & Radar) An airplane is flying (horizontally) at the height of 6 km on a flight path that will take it directly over a radar tracking station (on ground). If the distance D between the plane and the radar is decreasing at … 
Caculas
(Airplane & Radar) An airplane is flying (horizontally) at the height of 6 km on a flight path that will take it directly over a radar tracking station (on ground). If the distance D between the plane and the radar is decreasing at … 
calculus
A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station. … 
biology
9. A certain radar installation transmits electromagnetic radiation with a wavelength of 2.5 cm. What is the frequency of this radiation? 
Math
An airplane is flying at an altitude of 6 miles and passes directly over a radar antenna. When the plane is 10 miles away, the radar detects that the distance is changing at a rate of 240 mph. what is the speed of the plane? 
CALCULUS
A plane, flying horizontally at an altitude of 1 km and a speed of 500 km/hr, passes directly over a radar station. find the rate in km/hr at which the distance from the plane to the station is changing when it is 2 km away from the … 
physics
A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 118 km away. The radar antenna is pointing upward at an angle of 58.2 ° above the ground. Find (a) the x and … 
physics
A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be 181 km away. The radar antenna is pointing upward at an angle of 66.2 ° above the ground. Find (a) the x and …