Create an image illustrating a geographical scenario. There's a small, tranquil lake surrounded by lush greenery. To the one side of the lake, there's a charming gazebo with white marble pillars and a wooden pavilion. At the other side of the lake, there's a simple wooden dock protruding into the water. Also, somewhere 100 meters away from the gazebo there's a tall mature tree. The bearing from the gazebo to the dock is towards S 41° W. From the tree, the bearings to the gazebo and the dock are towards S 74° E and S 28° E, respectively. Please do not include any text in this illustration.

A bridge is to be built across a small lake from a gazebo to a dock. The bearing from the gazebo to the dock is S 41° W. From a tree x = 100 meters from the gazebo, the bearings to the gazebo and the dock are S 74° E and S 28° E, respectively (see figure in the link below). Find the distance from the gazebo to the dock. (Round your answer to the nearest whole number.)

www.webassign.net/larprecalcaga5/7-1-028-alt.gif

my answer is

(100sin46)/sin69 = 77.1m
is my answer correct?

distance TD = 100 m

GTD = 74-28 = 46 deg
TDG = 28 + 41 = 69 deg
TGD = 180 - 74-41 = 65 deg
now is 46 + 69 + 65 = 180? Yes, remarkable we have a triangle
so
sin 46/GD = sin 65/100

yes

I did the wrong leg

hey damon thanks,

can you help me on this question:
A plane flies 505 kilometers with a bearing of 316° (clockwise from north) from Naples to Elgin. The plane then flies 770 kilometers from Elgin to Canton (see figure in the link below). Canton is due west of Naples. Find the bearing of the flight from Elgin to Canton. (Round to the nearest whole number.)
www.webassign.net/larpcalclim2/6-1-048-alt.gif

To solve this problem, we can break it down into two smaller triangles: the triangle formed by the tree, gazebo, and dock, and the triangle formed by the tree, gazebo, and the point where the bridge meets the lake.

First, let's find the distance from the gazebo to the point where the bridge meets the lake.

Using the bearings provided, we know that the angle between the tree-gazebo bearing (S 74° E) and the tree-dock bearing (S 28° E) is 74° - 28° = 46°.

We also know that the angle between the tree-gazebo bearing (S 74° E) and the bridge-gazebo bearing (S 41° W) is 180° - 74° - 41° = 65°.

Now we can use the Law of Sines to find the distance from the gazebo to the point where the bridge meets the lake. Let's call this distance "d".

sin(65°) / 100 = sin(46°) / d

To find "d", we can rearrange the equation:

d = (100 * sin(65°)) / sin(46°)

Using a calculator, this gives us d ≈ 111.18 meters (rounded to two decimal places).

Now we need to find the distance from the point where the bridge meets the lake to the dock.

To do this, we can use the Law of Sines again with the triangle formed by the tree, gazebo, and dock.

Using the bearing of S 41° W, we can find the angle between the bridge-gazebo bearing and the tree-dock bearing:

180° - 41° = 139°

Let's call the distance from the point where the bridge meets the lake to the dock "x".

sin(139°) / 111.18 = sin(46°) / x

To find "x", we can rearrange the equation:

x = (111.18 * sin(46°)) / sin(139°)

Using a calculator, this gives us x ≈ 48.19 meters (rounded to two decimal places).

Finally, to find the total distance from the gazebo to the dock, we can add the distance from the gazebo to the point where the bridge meets the lake (d ≈ 111.18 meters) and the distance from the point where the bridge meets the lake to the dock (x ≈ 48.19 meters):

Total distance = 111.18 + 48.19 ≈ 159 meters (rounded to the nearest whole number).

Therefore, the distance from the gazebo to the dock is approximately 159 meters.

I have angle at G = 65

You have 69 which I had at the dock

Whoops sorry I did the tree to the dock

so (100sin46)/sin69 = 77.1 is correct?

sin 46/GD = sin 65/100

=(100sin46)/sin65=79.37
is this correct?