The bearing of a school from the sickbay is 310°.From the commandant's office ,317m due east of the sickbay,the bearing of the school library is 288°.How far is the library from the sickbay?
To find the distance between the library and the sickbay, we can use trigonometry and a bit of geometry. Let's break down the problem step by step:
Step 1: Understand the given information.
The bearing of the school from the sickbay is 310°.
The bearing of the library from the office is 288°.
The office is 317m due east of the sickbay.
Step 2: Draw a diagram.
Draw a simple diagram representing the situation. Place the sickbay at the origin (point O). Then, draw a line segment from O due east representing the office. Label the endpoint of this line segment as point E. Finally, draw a line segment starting from E with a bearing of 288°, representing the direction of the library. Label the endpoint of this line segment as point L.
(O)
/ |
/ | 317m
/ |
/ (E)
/ |
/ |
(L)
Step 3: Determine the angle between the library and the office.
Since the bearing of the library from the office is 288°, and the bearing is measured clockwise from the north, we need to find the angle between the line segment OE and the line segment EL. To do this, we subtract the given bearing from 360°.
Angle EOL = 360° - 288° = 72°
Step 4: Use the given information to form a triangle.
We can now form a right triangle using the line segments OE, EL, and OL. We are interested in finding the length of segment OL, which represents the distance between the library and the sickbay.
Step 5: Use trigonometry to find the length of OL.
In the right triangle OEL, we have one known side (OE = 317m) and one known angle (angle EOL = 72°). We can use the trigonometric function tangent (tan) to find the length of OL.
tan(angle EOL) = opposite / adjacent
tan(72°) = OL / OE
Let's solve for OL:
OL = OE * tan(angle EOL)
OL = 317m * tan(72°)
Using a calculator, we can calculate the value of OL:
OL ≈ 1279.81m
Therefore, the library is approximately 1279.81 meters away from the sickbay.
Draw the diagram.
if the school is x meters west and y meters north of the sickbay, and z meters from the office, we have
y/x = tan30°
y/(x+317) = tan 18°
so, x tan30° = (x+317) tan18°
and, now that you have x,
(x+317)/z = cos 18°
So crank it out.