a pilot entering a bay on course 60.3 degrees at the speed of 12 knots sees a light bearing 37.3 degrees true, and 20 mins later he sees it bearing 20 degrees true. if he keeps the same course and speed,(a)when will he be nearest to the lighthouse and how near (b) when will he be 11 miles from the lighthouse?
If I understand true bearings as relating to true north, then we have a diagram where
L is the light
A is the initial observation point
B is the second observation point
In 20 minutes, the ship has traveled 4 miles. AB=4
∠A = 60.3-37.3 = 23°
∠B = 180-(60.3-20) = 139.7°
∠L = 37.3-20 = 17.3°
Using the x-y plane, as opposed to compass bearings, if the first sighting was taken at (0,0), then
the ship's position is
s = tan29.7° x = 0.57x
the light is somewhere along the line
y = tan52.7° x = 1.31x
Now, after 20 minutes, s is 4 miles from (0,0), at (3.47,1.98)
So, the second sighting is somewhere along the line
y-1.98 = tan70° (x-3.47), or
y = 2.74(x-3.47)+1.98
y = 2.74x - 7.53
That places the light at (5.27,6.90)
At time t hours, we have the coordinates of s as
x = 12*cos29.7 t = 10.42t
y = 12*sin29.7 t = 5.95t
Now, the distance from the ship to the light at time t is
d^2 = (10.42t-5.27)^2 + (5.95t-6.90)^2
d^2 = 143.98t^2 - 191.94t + 75.39
When is that a minimum?
dd/dt = (143.98t-95.97)/d
Since d≠0, dd/dt=0 when t = .666, or 40 minutes from the first sighting.
Extra credit: how close is the nearest approach? 3.38 miles.
when is the ship 11 miles from the light? when
143.98t^2 - 191.94t + 75.39 = 11^2
t = 1.54 hours
Aha! So, the pilot is on a mission to find the lighthouse, huh? Let's navigate through this question with a touch of humor, shall we?
(a) To find out when the pilot will be nearest to the lighthouse, we need to use our trusty sense of direction and some simple calculations. Now, don't worry, I won't bore you with all the nautical mumbo-jumbo.
Let's assume that the pilot is only focusing on finding the nearest restroom (a crucial matter, I assure you). Based on the information provided, we can deduce that he'll be closest to the lighthouse when he sees it bearing 20 degrees true. So, let's set our course for some laughter and continue!
Now, back to business. If the pilot sees the lighthouse bearing 20 degrees true after 20 minutes, we can determine that his rate of change of bearing is -17.3 degrees per hour. To calculate when he'll be nearest to the lighthouse, we'll assume that the rate of change remains constant. Stay with me now!
To find the time when the rate of change equals zero, we can solve for t in the equation: 37.3 - 17.3t = 20, where t is time in hours. When we crunch the numbers (not popcorn, unfortunately), we find t ≈ 1.09 hours. But let's not forget the 20 minutes! Convert that into hours, and we have 20/60 ≈ 0.33 hours.
So, when we add 1.09 hours and 0.33 hours, we get approximately 1.42 hours. Voilà! That's the time when our pilot will be closest to the lighthouse. As for the distance, let's say they'll be within "whispering distance." I'd say that's around 1-2 miles. Those faint signals, you know?
(b) Now, let's tackle the second part of the question. When will our pilot be 11 miles from the lighthouse? Oh boy, we're going on a journey! Buckle up!
Since we know the speed of our fearless pilot is 12 knots (speedy, huh?), we can set up a little equation: D = 12t, where D is the distance in nautical miles, and t is time in hours. We want D to be 11 miles, so we have 11 = 12t. Solving for t, we find t = 11/12 ≈ 0.92 hours.
Now, let's not forget those 20 minutes we mentioned earlier. Convert that into hours, and we have 0.33 hours. So, when we add 0.92 hours and 0.33 hours, we get approximately 1.25 hours. That's when our pilot will be 11 miles from the lighthouse! Just far enough to see the light, but too close to borrow sugar.
Well, there you have it! Our pilot will be closest to the lighthouse in a charmingly brief amount of time and at a distance that won't require shouting. Safe travels, aviator! And don't forget to bring back a souvenir from the lighthouse gift shop!
To determine when the pilot will be nearest to the lighthouse and how near, we can analyze the scenario step by step.
Step 1: Calculate the time it takes for the pilot to change his bearing by 37.3 - 20 = 17.3 degrees true over 20 minutes.
To find the rate of change in bearing, divide the angle change by the time taken:
Rate of change in bearing = (17.3 degrees true) / (20 minutes)
Step 2: Calculate the distance traveled by the pilot in the 20-minute interval.
To find the distance traveled, multiply the speed (12 knots) by the time (20 minutes):
Distance traveled = (12 knots) * (20 minutes)
Step 3: Use the information to calculate the rate at which the pilot is closing in on the lighthouse.
The rate of closure can be determined by multiplying the speed (12 knots) by the sine of the angle between the pilot's course and the bearing to the lighthouse at any given time.
Rate of closure = (12 knots) * sin(angle between course and bearing)
Step 4: Calculate when the pilot will be nearest to the lighthouse and how near.
The pilot will be nearest to the lighthouse when the rate of closure is at its maximum. To calculate when this occurs, take the derivative of the rate of closure with respect to the angle.
Now let's put these steps into practice:
Step 1: Calculate the rate of change in bearing:
Rate of change in bearing = 17.3 degrees true / 20 minutes = 0.865 degrees/min
Step 2: Calculate the distance traveled:
Distance traveled = 12 knots * 20 minutes = 240 nautical miles
Step 3: Calculate the rate of closure:
Rate of closure = 12 knots * sin(37.3 - 60.3) degrees = -8.740 knots
Step 4: Calculate when the pilot will be nearest and how near:
To calculate when the pilot will be nearest, we need to find the time at which the rate of closure is maximum. Since the rate of closure is negative, it will have the highest magnitude (closest to zero) when it changes from negative to positive. This occurs when the bearing is equal to the course, i.e., when the angle between the course and the bearing is 0 degrees.
Let's calculate the angle when the rate of closure is maximum:
Maximum rate of closure = 12 knots * sin(0) degrees = 0 knots
Now, let's calculate the time it takes for the bearing to change by 0.865 degrees/min:
Time for bearing change = (0 degrees - 37.3 degrees) / (0.865 degrees/min) = -37.3 / 0.865 minutes
Since time cannot be negative, we take the absolute value and convert it to positive:
Time for bearing change = 37.3 / 0.865 minutes ≈ 43.09 minutes
Therefore, the pilot will be nearest to the lighthouse in approximately 43.09 minutes.
To calculate how near the pilot will be, we use the rate of closure when the bearing is 0 degrees:
Rate of closure at nearest approach = 12 knots * sin(0) degrees = 0 knots
Therefore, when the pilot is nearest to the lighthouse, the distance between them will not change, and the pilot will be as near as they were at that point.
Now, let's move on to the second part of the question:
To calculate when the pilot will be 11 miles from the lighthouse, we can use the rate of closure calculated earlier.
Distance traveled = 240 nautical miles
Rate of closure = -8.740 knots
Time taken to travel 11 miles = 11 miles / (-8.740 knots) ≈ -1.26 hours
Since time cannot be negative, we take the absolute value:
Time taken to travel 11 miles ≈ 1.26 hours
Therefore, the pilot will be 11 miles from the lighthouse in approximately 1.26 hours.
To solve this navigation problem, we can use trigonometry and basic geometry. Let's break down the steps to find the answers.
First, let's draw a diagram to visualize the situation. Draw a horizontal line to represent the course of the pilot entering the bay. At the left end of the line, draw a dot to represent the pilot's starting point. At some point along this line, draw another dot to represent the position of the light.
Using the given information, we know the initial bearing from the pilot to the light is 37.3 degrees true. We also know that 20 minutes later, the bearing from the pilot to the light is 20 degrees true.
Step 1: Finding the time when the pilot is nearest to the lighthouse:
To determine when the pilot is closest to the lighthouse, we need to find the point along the path where the distance to the lighthouse is the shortest. This will occur when the perpendicular distance from the pilot's path to the line connecting the pilot's starting point and the lighthouse is minimized.
Since the pilot maintains the same course and speed, the line connecting the pilot's starting point and the lighthouse is constant. Therefore, we can focus on finding the point where the perpendicular distance is minimized.
To find this point, we can use trigonometry. Note that the initial bearing of the pilot (60.3 degrees) is the angle between the pilot's course and the line connecting the starting point and the lighthouse.
Step 2: Finding the distance when the pilot is nearest to the lighthouse:
Once we have determined the time when the pilot is nearest to the lighthouse (from step 1), we can calculate the distance by using the pilot's speed. The distance covered by the pilot is given by the formula:
Distance = Speed × Time
Step 3: Finding the time when the pilot is 11 miles from the lighthouse:
To calculate the time when the pilot is 11 miles from the lighthouse, we can use the same formula as in step 2:
Time = Distance / Speed
Substitute the known distance (11 miles) into the formula, and calculate the time it takes to cover this distance at the given speed.
By following these steps, you should be able to determine when the pilot is nearest to the lighthouse and how near, as well as the time when the pilot is 11 miles away from the lighthouse.