1. A boat sails for 4 hours at 35 mph in a direction 135degrees 51'. how far shouth has it sailed (to the nearest mile)?
Is it 96 miles??
2. A ship travels 55 km on a bearing of 11degrees, and then travels on a bearing of 101degrees for 133 km. find the distance of the end of the trip from the starting point to the nearest kilometer.
Im not sure about this one!
On the assumption that you consider NORTH to have a bearing of 0 and 135 is a clockwise rotation from there, you would have a triangle with the x-axis as one side, the hypotenuse as 140 miles and you want the vertical side of that triangle,
so sin 45 = x/140
x = 140/sin45
= 198
for the second one, after making a sketch I found the angle between the two paths to be 90 degrees, so we want to find the hypotenuse of a right-angled triangle with sides 55 and 133
h^2 = 55^2 + 133^2
h = 143.9
1. To find the distance the boat has sailed, we can use the distance formula:
Distance = Speed × Time
In this case, the speed is 35 mph and the time is 4 hours. So,
Distance = 35 mph × 4 hours
Distance = 140 miles
Therefore, the boat has sailed 140 miles.
2. To find the distance from the starting point to the end of the trip, we can use the Law of Cosines. The Law of Cosines states that:
c² = a² + b² - 2ab * cos(C)
In this case, we have two sides and the included angle (SAS), so we can use the formula:
c² = a² + b² - 2ab * cos(C)
c² = 55² + 133² - 2 * 55 * 133 * cos(101° - 11°)
c ≈ √(55² + 133² - 2 * 55 * 133 * cos(90°))
Calculating this expression gives us:
c ≈ √(55² + 133² - 2 * 55 * 133 * (-0.0871557))
c ≈ √(3025 + 17689 + 12267.70)
c ≈ √(32981.70)
c ≈ 181.57 km
Therefore, the distance from the starting point to the end of the trip is approximately 181.57 km (rounded to the nearest kilometer).
To solve the first question, you can use trigonometry and the concept of vectors. The boat is sailing at a constant speed for 4 hours, so you can find the distance it has traveled by multiplying the speed (35 mph) by the time (4 hours). This gives you 140 miles.
To determine how far south the boat has traveled, you need to consider the angle it is sailing at. The given direction is 135 degrees 51'. However, the angle should be converted to radians for further calculations.
To convert from degrees to radians, use the formula:
radians = degrees * (π/180)
So, the angle in radians is:
135 degrees 51' * (π/180) = 2.3719 radians
Now, you can use trigonometry to find the southward distance traveled by the boat. Since the south direction is opposite to the angle, you can use the cosine function. The formula is:
south distance = distance * cos(angle)
Plugging in the values:
south distance = 140 miles * cos(2.3719 radians) ≈ 140 miles * (-0.7308) ≈ -102.312 miles
Since the question asks for the distance to the nearest mile, we can round the answer to the nearest whole number. Thus, the boat has sailed approximately 102 miles south.
Regarding the second question, you can use the concept of vectors and the law of cosines to solve it. Draw a diagram or visualize the ship's movement.
Given:
First leg: 55 km, bearing 11 degrees
Second leg: 133 km, bearing 101 degrees
To find the distance of the end of the trip from the starting point, you need to find the resulting displacement vector using the two legs. The displacement represents the straight-line distance from the starting point to the end point.
To calculate the x-component (horizontal) of the displacement vector:
x = 55 km * cos(11 degrees) + 133 km * cos(101 degrees)
To calculate the y-component (vertical) of the displacement vector:
y = 55 km * sin(11 degrees) + 133 km * sin(101 degrees)
Using the law of cosines, you can find the magnitude of the displacement vector (d):
d = √(x^2 + y^2)
Calculate x and y, then substitute them into the magnitude formula to find the distance from the starting point to the end point. Once you have the value of d, round it to the nearest kilometer to get the final answer.