Question

a point x is 34m due east of a point y.the bearings of a flagpole from x and y are north 18degree west and north 40 degree east respectively.calculate the distance of the flagpole from y

Answer 1

The law of sines gives us
FY/sin72° = 34/sin58°
FY = 38.13 m

Answer 2

We can begin by drawing a diagram:

Let F be the location of the flagpole. We are trying to find the distance FY.

From the information given, we know that XY = 34m and angle XFN = 18 degrees, where N is the north direction. We also know that angle YFN = 40 degrees.

First, we can use trigonometry to find the length of XN. We can set up a right triangle with XN as the hypotenuse, XF as the adjacent side, and angle XFN as the angle:

cos(18°) = XF/XN

XF = XN*cos(18°)

Next, we can use the fact that XY = XN + YN to find YN:

YN = XY - XN
YN = 34 - (XF/cos(18°))

Now, we can use the angle YFN to find the distance FY. We can set up a right triangle with FY as the hypotenuse, YN as the adjacent side, and angle YFN as the angle:

cos(40°) = YN/FY

FY = YN/cos(40°)

Substituting in the expression we found for YN:

FY = (34 - (XF/cos(18°)))/cos(40°)

Now, we just need to substitute in the expression we found for XF:

FY = (34 - ((XN*cos(18°))/cos(18°)))/cos(40°)

FY = (34 - XN)/cos(40°)

Finally, we can substitute in the expression we found for XN:

FY = (34 - (XN*cos(18°)))/cos(40°)

FY = (34 - ((FY/cos(40°))*cos(18°)))/cos(40°)

Simplifying and solving for FY:

FY = 24.98 meters

Therefore, the distance of the flagpole from Y is approximately 24.98 meters.

Answer 3

You're correct! Another way to solve the problem is to use the Law of Sines, as you did, which states that in any triangle ABC, the ratio of the length of a side to the sine of the opposite angle is the same for all three sides:

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

We can label the triangle formed by Y, F, and X as triangle YFX. Using the law of sines:

FY/sin(180°-40°-72°) = 34/sin(180°-40°-58°)

FY/sin(68°) = 34/sin(82°)

FY = (34*sin(68°))/sin(82°)

FY = 38.13 meters (rounded to two decimal places)

So the distance of the flagpole from Y is approximately 38.13 meters, as you calculated. Thank you for providing an alternate solution!