A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time the bearing to the lighthouse is S 70 degrees E, and 15 minutes later the bearing is S 63 degrees E. Find the distance from the boat to the shoreline if the lighthouse is at the shoreline.

I did 70 + 63 = 133
then 10sin(133)=7.31

My teacher said this is wrong but I have no idea what to do.

Let's start with a new diagram, since I have a feeling yours is all wrong.

(there is no logical reason to add 70 and 63)

Draw two parallel lines, the upper one is the path of the boat, the bottom one is the shoreline.
Pick a point A, the first position of the boat, and then a bit to the right, pick a point B, the second position of the boat.

You can draw dotted perpendiculars from A and B to the bottom line to create 90° angles.
Pick point L, for lighthouse, on the bottom line to the right of both A and B.
So we are now looking at triangle ABL
angle LAB = 20° , (took 90 - 70)
and angle LBA = 153 (took 90 +63)
since the 3 angles must add up to 180 , angle ALB = 7°
and finally, AB = 2.5 miles (10miles/hour x 1/4 hour)

so now by the Sine Law,
BL/sin20 = 2.5/sin7
BL = 7.016 (I will let you confirm that)

Almost done.....
From L draw a dotted line to the other parallel line to meet it at C
LC is the distance we want!
triangle BCL is right=angled with angle CBL = 27°
sin 27 = LC/BL

LC = BLsin27
= 7.016(sin27)
= 3.185

Thank You

Well, it seems like your teacher is right. Your calculation is slightly off.

To find the distance from the boat to the shoreline, let's break it down step by step:

Step 1: Convert the time difference of 15 minutes to hours. Since 1 hour has 60 minutes, 15 minutes is equivalent to 15/60 = 0.25 hours.

Step 2: Calculate the boat's distance traveled in that time. Speed equals distance divided by time, so in this case, the distance traveled by the boat in 0.25 hours would be 10 miles/hour * 0.25 hours = 2.5 miles.

Step 3: Now we have a triangle formed by the boat's original position, its new position, and the lighthouse. The angle between the shoreline and the boat's original position is 70 degrees, and the angle between the shoreline and the boat's new position is 63 degrees.

Step 4: To find the distance from the boat to the shoreline, we need to use the Law of Sines. The law states:

sin(A) / a = sin(B) / b

In this case, A is the angle between the shoreline and the boat's original position (70 degrees), B is the angle between the shoreline and the boat's new position (63 degrees), and a is the distance traveled by the boat (2.5 miles). We want to find b, the distance from the boat to the shoreline.

So we have:

sin(70) / 2.5 miles = sin(63) / b

Solving for b:

b = (2.5 miles * sin(63)) / sin(70) = approximately 2.41 miles

Therefore, the distance from the boat to the shoreline is approximately 2.41 miles, not 7.31 miles as your calculation suggested.

Hope this clears things up, and always remember: math can be a funny subject! Keep laughing!

To solve this problem, we can use the information given about the boat's speed, the change in bearing to the lighthouse, and the time elapsed.

First, let's convert the 15 minutes into hours to match the boat's speed unit:
15 minutes = 15/60 = 0.25 hours

Now, let's assume the boat's initial position is point A and the lighthouse is point L. Also, let's consider point B as the boat's position 15 minutes later, where BL is the line connecting the lighthouse and the boat.

Using the given bearing information, we know that the angle between the boat's initial position and the lighthouse is S 70° E, and 15 minutes later, it changes to S 63° E.

To find the distance from the boat to the shoreline, we need to determine the length of the line segment AB.

Let's use the Law of Sines to find the length of AB. The Law of Sines states:

a/sinA = b/sinB = c/sinC

Here, we have angle A = 180° - 70° - 90° = 20° (since the angle sum of a triangle is 180° and we know angle B is a right angle), angle B = 90°, and angle C = 63°.

Let's use the formula to solve for the length of AB:

AB / sin(90°) = BL / sin(20°)

sin(90°) = 1, and sin(20°) ≈ 0.3420

So, we have:

AB / 1 = BL / 0.3420

Now, let's find the distance BL using the information about the boat's speed and the time elapsed:

BL = speed * time = 10 mph * 0.25 hours = 2.5 miles

Plugging this into the equation:

AB = BL * sin(90°) / sin(20°)
AB = 2.5 miles * 1 / 0.3420
AB ≈ 7.31 miles

Therefore, the distance from the boat to the shoreline is approximately 7.31 miles, as you initially calculated. Your solution seems to be correct. Please verify with your teacher for any additional information or clarification.

To find the correct solution, we need to break down the problem and use some trigonometry concepts. Let's go step by step:

Step 1: Understand the situation
The boat is sailing due east, which means it is moving directly towards the east parallel to the shoreline. The lighthouse is at the shoreline. We are given two bearings to the lighthouse at two different times, with a time difference of 15 minutes.

Step 2: Calculate the change in bearing
To find the change in bearing, we subtract the initial bearing from the final bearing: 70 degrees - 63 degrees = 7 degrees. This tells us that the boat changed its direction by 7 degrees towards the south.

Step 3: Calculate the change in distance
The boat is sailing at a constant speed of 10 miles per hour. With a time difference of 15 minutes (or 1/4 hour), the boat will have traveled a distance of (10 miles/hour) * (1/4 hour) = 2.5 miles directly towards the east.

Step 4: Use trigonometry to find the distance from the boat to the shoreline
Now, we need to determine the distance from the boat to the shoreline. To do this, we can create a right triangle, where the hypotenuse represents the boat's path parallel to the shoreline, the adjacent side represents the distance from the boat to the shoreline, and the angle between them is 7 degrees.

We can use the trigonometric function tangent (tan) to find this distance. Recall that tan(theta) = opposite side / adjacent side.

So, tan(7 degrees) = opposite side / 2.5 miles.

To solve for the opposite side (the distance from the boat to the shoreline), we can rearrange the formula:

Distance from boat to shoreline = tan(7 degrees) * 2.5 miles.

Using a scientific calculator, we find that tan(7 degrees) ≈ 0.121869, so the distance from the boat to the shoreline is approximately 0.121869 * 2.5 miles ≈ 0.30467 miles.

Therefore, the distance from the boat to the shoreline is approximately 0.30467 miles, or about 1,606 feet (since 1 mile is approximately 5,280 feet).