Two patrol boats M3 and M7 leave port at the same port at the same time. M3 heads due West and M7 on the bearing 277 degrees. After 30 minutes M7 has travelled 18 meters and observe M3 in the direction due south.
a. How far is M3 from M7?
b. How far has M3 travelled?
a. We can use trigonometry to find the distance between M3 and M7. Let's call this distance "x". We know that M7 has traveled 18 meters in 30 minutes, which means its speed is:
18 meters / 30 minutes = 0.6 meters per minute
We also know that M7 and M3 are forming a right angle (M7 is heading on a bearing of 277 degrees, which is 13 degrees from due south). So we can use the tangent function to find x:
tan(13) = x / 0.6
x = 0.6 * tan(13) ≈ 0.13 km
Therefore, M3 is about 0.13 kilometers away from M7.
b. Since M3 has been heading due West for 30 minutes, we can find the distance it has traveled by multiplying its speed by the time:
distance = speed * time
We don't know M3's exact speed, but we do know that it has been traveling in a straight line. So we can use the Pythagorean theorem to find the distance it has traveled:
distance = √(x^2 + (18/60)^2)
where x is the distance we found in part (a), and 18/60 is the distance that M7 has traveled (since speed = distance / time, and M7 has traveled for 30 minutes).
Plugging in the numbers, we get:
distance = √(0.13^2 + 0.005^2) ≈ 0.13 km
Therefore, M3 has traveled about 0.13 kilometers.
To solve this problem, we can use trigonometry and basic distance-speed-time formulas.
a. How far is M3 from M7?
Since M7 has traveled for 30 minutes and observed M3 in the direction due south, we can form a right-angled triangle with the distance traveled by M7 (18 meters) as the base and the distance between M3 and M7 as the hypotenuse. We need to find the length of the hypotenuse.
Let's denote the distance between M3 and M7 as "x" meters. We can use trigonometry to find x.
First, we find the length of the adjacent side of the triangle (the distance traveled by M3) using the formula:
Adjacent length = 18 meters * cos(90 – 277) [converting the bearing to an angle in degrees]
Adjacent length = 18 meters * cos(-187) [since 90 - 277 = -187]
Next, we can use Pythagoras' theorem to find the hypotenuse length (x) using the formula:
x^2 = Adjacent length^2 + opposite length^2
Since the opposite length is the same as the distance traveled by M7 (18 meters), the equation becomes:
x^2 = (18 meters * cos(-187))^2 + (18 meters)^2
Simplifying this equation, we find:
x^2 = (18 meters)^2 * [cos(-187))^2 + 1]
x^2 = 324 * [cos(-187))^2 + 1]
Now, we need to calculate cos(-187):
cos(-187) = cos(360 - 187) [Using the periodicity property of cosine]
cos(-187) = cos(173)
Using a scientific calculator or trigonometric tables, we find:
cos(173) ≈ -0.996
Substituting this value back into the equation, we have:
x^2 = 324 * [(-0.996)^2 + 1]
x^2 = 324 * (0.992 + 1)
x^2 = 324 * 1.992
x^2 ≈ 646.208
Taking the square root of both sides, we find:
x ≈ √646.208
x ≈ 25.41 meters
Therefore, M3 is approximately 25.41 meters away from M7.
b. How far has M3 traveled?
M3 heads due West, and since it has traveled for the same amount of time as M7 (30 minutes), we can calculate its distance using the formula:
Distance = Speed * Time
As M3 is traveling due West, we don't have the direction or angle given. So, we can consider it as a simple 1-dimensional motion.
Given that M3 and M7 left port at the same time, M3's distance traveled is equal to M7's distance traveled.
Therefore, M3 has traveled 18 meters as well.
To answer both questions, we need to find the distances using the information given.
a. How far is M3 from M7?
To find the distance between M3 and M7 at the given time, we can visualize a triangle formed by the two patrol boats and their starting point. We can use the concept of trigonometry to find the distance between M3 and M7.
Step 1: Draw the diagram:
Consider a right-angled triangle, with M7 at the right angle, M3 at the top vertex, and the starting point at the bottom vertex. The distance between M7 and the starting point is 18 meters.
M7
/|
/ |
/ |
M3 / |
/ |
/ |
/ |
Starting point (right angle)
Step 2: Calculate the distance between M7 and M3 using trigonometry:
We have the angle opposite to the side we want to find, which is the angle M3M7S (where S is the starting point). The angle is 90 degrees - 277 degrees = -187 degrees. Since negative angles can be dealt with as positive angles, we consider the positive angle: 187 degrees.
Now, using the tangent function, we can calculate the distance between M7 and M3:
tan(187 degrees) = (M3M7) / (M7S)
We know that M7S is 18 meters, so we can rearrange the formula to solve for M3M7:
(M3M7) = tan(187 degrees) * 18
Now, we can use a scientific calculator to find the value of tan(187 degrees) * 18.
Calculating: tan(187 degrees) * 18 ≈ -15.47
Therefore, M3 is approximately 15.47 meters away from M7.
b. How far has M3 traveled?
Given that M3 traveled due West, we can use the formula for distance = speed x time.
We know the time M3 traveled is 30 minutes, which is equal to 30/60 = 0.5 hours.
We can use the formula:
Distance = Speed x Time
Assuming the speed of M3 is constant, we can solve the equation. Unfortunately, the speed of M3 is not provided, so we cannot calculate the exact distance traveled.