Lighthouse B is 7 miles west sof lighthouse A. A boat leaves A and sails 15 miles. At this time, it is sighted from B. If the bearing of the boat from B is N62degE, how far from B is the boat?
To find the distance from B to the boat, we can use trigonometry. Let's assume that the boat is at point C.
First, let's draw a diagram to visualize the situation:
C
/|
/ |
/ |15 miles
/ |
B/____|____A
We know that Lighthouse B is 7 miles west of Lighthouse A. So, the distance between B and A is 7 miles.
From the information given, the bearing of the boat from B is N62°E.
This means that if we draw a line from B to the boat, the angle between this line and the north direction is 62°, and the angle between this line and the east direction is 90° - 62° = 28°.
Considering the triangle BAC, where AC is the distance from B to the boat, we can determine the distance AC using trigonometry.
Using the cosine rule, we have:
cos(28°) = 7 miles / AC
Rearranging the equation, we can solve for AC:
AC = 7 miles / cos(28°)
Using a calculator, we find:
AC ≈ 7 miles / (cos 28°) ≈ 7.98 miles
Therefore, the boat is approximately 7.98 miles away from Lighthouse B.
To solve this problem, we can use the concept of trigonometry and bearings.
Let's break down the information provided:
- Lighthouse B is 7 miles west of lighthouse A.
- The boat leaves lighthouse A and sails 15 miles.
- At this time, it is sighted from lighthouse B.
- The bearing of the boat from B is N62degE.
Now, let's visualize the situation. Draw a diagram with lighthouses A and B, with B located to the west of A. Place the boat somewhere in between the two lighthouses, after sailing 15 miles from A.
To find the distance of the boat from lighthouse B, we can divide the problem into two right-angled triangles: one with lighthouse A, the boat, and the point where it is sighted from B, and another with lighthouse B, the boat, and the same point.
In triangle ABO (where O is the point where the boat is sighted), we have:
- Angle BAO = 62 degrees (given)
- AB (distance between A and B) = 7 miles (given)
- AO (distance sailed by the boat) = 15 miles (given)
Now, we can use trigonometry to find the distance BO (the distance of the boat from lighthouse B).
Using the tangent function, we have:
tan(BAO) = BO / AB
Rearranging the equation:
BO = AB * tan(BAO)
Plugging in the values we know:
BO = 7 miles * tan(62 degrees)
To find the value of tan(62 degrees), use a scientific calculator or a trigonometric table. After calculating the tangent value, you can multiply it by 7 miles to find the distance BO.
So, the final step is to calculate BO using the above formula to find how far the boat is from lighthouse B.
Y = 15*sin(90-62) = 7.04 mi.
X = 7 + 15*cos(90-62) = 20.2 mi.
d = sqrt(20.2^2+7.04^2) = 21.4 mi.