A boat sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time the bearing to the light house is S70 degrees E, and 30 minutes later the bearing is S63 degrees E. Find the distance from the boat to the shoreline if the lighthouse is at the shoreline.
To find the distance from the boat to the shoreline, we can use the concept of a triangle formed by the boat, the lighthouse, and a point on the shoreline.
First, we need to determine the boat's velocity components in the eastward (horizontal) and northward (vertical) directions. Since the boat is sailing due east, its velocity component in the eastward direction is 10 mph. Since the boat is sailing parallel to the shoreline, its velocity component in the northward direction is zero.
Next, we can consider the two positions of the boat and the corresponding bearings to the lighthouse. Drawing a diagram can help visualize this situation.
Let's assume the boat starts at point A at the given time, and 30 minutes later it is at point B. The bearing to the lighthouse at point A is S70 degrees E, and at point B, it is S63 degrees E.
To find the distance from the boat to the shoreline, we can determine the distance between points A and B, and then subtract the distance between point B and the shoreline.
Since the boat's eastward velocity is constant at 10 mph, the distance from A to B can be calculated by multiplying the velocity by the time interval:
Distance AB = Velocity * Time Interval
= 10 mph * 0.5 hours (30 minutes expressed in hours)
= 5 miles
Now, we need to calculate the distance from point B to the shoreline. From the given bearing of S63 degrees E at point B, we know that the boat is moving at an angle of 63 degrees with respect to the eastward direction.
Using trigonometry, we can determine the distance from the boat to the shoreline by calculating the horizontal (eastward) displacement between point B and the shoreline.
Let's assume the horizontal displacement from point B to the shoreline is x miles.
Using the sine function, we can write the following equation:
sin(63 degrees) = x / 5 (since the vertical displacement is zero)
Rearranging the equation, we get:
x = 5 * sin(63 degrees)
x ≈ 4.3 miles (approximately)
Therefore, the distance from the boat to the shoreline is approximately 4.3 miles.
the boat has traveled 5 miles. So, if the distance to shore is d,
d tan70° - d tan63° = 5