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 co-ordinate 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 y-intercept 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 x-axis 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)
x-intercept 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)
To solve this problem, we need to find the point where the tugboat intersects the radar region. We can break down the problem into two parts:
Part 1: Find the intersection point
1. Start by drawing a diagram to visualize the situation. Draw the radar buoy at the center and mark its radius of 12 miles.
2. Mark the starting location of the tugboat, which is 18 miles SOUTH and 10 miles EAST of the radar buoy.
3. Connect the starting location of the tugboat with the NORTHERNMOST point of the radar region. This line represents the tugboat's initial trajectory.
4. Extend the line connecting the tugboat's starting location with the NORTHERNMOST point of the radar region.
5. The point where this line intersects the radar region is the intersection point we are looking for.
Part 2: Calculate the time in the radar region
6. Calculate the distance between the starting location of the tugboat and the intersection point. In this case, it would be the hypotenuse of a right triangle with one leg of 18 miles (the SOUTH distance) and the other leg of 10 miles (the EAST distance).
Distance = √((18^2) + (10^2))
7. Now we need to calculate the time it takes for the tugboat to cover this distance. We know that the tugboat travels at a constant speed of 15 mph.
Time = Distance / Speed
8. Plug in the values and calculate the time in hours.