A ship at A is to sail to C, 56km north and 258km east of A. After sailing N25°10’E for 120mi to P, the ship is headed toward C. Find the distance of P from C and the required course to mean C.
Use the same method I showed you in your post before this.
The exception here is that you know the final destination to be the ordered pair (258,56)
After converting (120 , N25°10' E) subtract that vector from (258,56)
I am assuming your distance of 120 mi was meant to be 120 km.
BTW, for an alternate solution look at
http://www.jiskha.com/display.cgi?id=1357447890
where the same question was posted and answered before.
Are you the same person, by the exact typing error and use of "mean" for C I assume you are.
To find the distance from point P to point C, we can use the Pythagorean theorem. The ship sailed 56km north from point A, and then 258km east, which forms a right-angled triangle. Let's call the distance from point P to point C as x.
Using the Pythagorean theorem, we have:
x^2 = (258km)^2 + (56km)^2
Simplifying this equation:
x^2 = 66564km^2 + 3136km^2
x^2 = 69600km^2
Taking the square root of both sides:
x = 264km
Therefore, the distance from point P to point C is 264km.
Now, let's find the course required to reach point C. We know that the ship sailed N25°10’E for 120 miles.
To determine the required course to reach point C, we need to find the bearing, which is the angle measured clockwise from true north.
The bearing can be found using trigonometry. In this case, we have a right-angled triangle where the side opposite the desired angle is 56km (north) and the side adjacent to the desired angle is 258km (east).
Using the inverse tangent function (tan⁻¹), we can find the angle:
tan⁻¹(56km/258km) = 12.267°
So, the bearing to mean point C is approximately N12°16’E.
To find the distance of P from C and the required course to reach C, we can break down the problem into steps.
Step 1: Find the coordinates of C.
The ship is initially at A, and it sails 56 km north and 258 km east to reach C. Let's denote the coordinates of A as (0, 0).
The coordinates of C are therefore (258 km, 56 km).
Step 2: Determine the distance between P and C.
The ship sails N25°10’E for 120 mi to reach P. To calculate the distance between P and C, we will use the Pythagorean theorem.
First, we need to convert the angle from degrees, minutes, and seconds to decimal degrees.
N25°10’E can be converted to 25.167°.
Next, we can calculate the eastward (x) and northward (y) displacements from P to C.
x = 120 mi * cos(25.167°)
y = 120 mi * sin(25.167°)
Using the coordinates of C and the displacements, we can calculate the distance between P and C.
Distance (d) = √[(258 km - x)^2 + (56 km + y)^2]
Step 3: Find the required course to reach C.
The required course is the angle between the line connecting P and C and the eastward direction.
To find the required course, we can use the inverse tangent function:
Required course = atan2(y, x)
Now that we have the plan, let's compute the values.
Step 1:
Coordinates of C: (258 km, 56 km)
Step 2:
Convert the angle N25°10'E to decimal degrees:
25.167°
Calculate the eastward (x) and northward (y) displacements:
x = 120 mi * cos(25.167°)
y = 120 mi * sin(25.167°)
Calculate the distance (d) between P and C:
Distance (d) = √[(258 km - x)^2 + (56 km + y)^2]
Step 3:
Calculate the required course:
Required course = atan2(y, x)
By following these steps and performing the necessary calculations, you will be able to find the distance of P from C and the required course to reach C.