A tree is on bearing s 36° w from a point x and s 78°w from a point y. If x is 200m due east of y . calculate the distance of the tree from y to the nearest metre
In triangle XYT
<XYT=90-78=12
<YXT=90-36=54
<YTX=180-(12+54)=114(SUM OF ANGLES IN A TRIANGLE)
USING SINE RULE,
|YT|/Sin54=200/sin114
-|YT|=(200×sin 54)/sin 114
ANSWER:|YT|=177m
Well, well, well, it seems like we have a little geography puzzle here! Let's see if we can solve it together, shall we?
First things first, since we know that the tree is s 36° w from point x and s 78°w from point y, we can conclude that the angle between x and y is 78° - 36° = 42°.
Now, we also know that point x is 200m due east of point y. Since we have an angle between the two points and a distance (200m), we can use some good ol' trigonometry to find the distance of the tree from point y.
Using the sine function, we can set up the equation: sin(42°) = Opposite / Hypotenuse.
The hypotenuse, in this case, represents the distance between point y and the tree. So, we can rewrite the equation as: sin(42°) = Opposite / Distance.
Rearranging the equation, we find: Distance = Opposite / sin(42°).
Plugging in the values, we get: Distance = 200m / sin(42°). Now, we just need to whip out our calculators and do the math.
Calculating that, we find that the distance of the tree from point y is approximately 286 meters to the nearest meter.
So, drumroll please... the distance of the tree from point y is approximately 286 meters!
I hope that brings a little laughter and excitement to your math problem. If you have any more questions, feel free to ask!
To solve this problem, we can break it down into smaller steps:
Step 1: Draw a diagram illustrating the given information.
Let's draw a diagram to help visualize the situation. Let's represent point X as X, point Y as Y, and the tree as T. We can also draw the angle measurements.
X-----200m-----Y
| /
| /
| /
T
Step 2: Calculate the bearing angle between point X and the tree.
We know that the tree is on bearing S 36° W from point X. This means that the angle between the line connecting X and T and the south direction is 36°. However, we want to find the angle between the line connecting X and T and the east direction.
To find the angle between the east direction and the line connecting X and T, we subtract the given angle of 36° from 90° (since the east direction is perpendicular to the south direction).
Angle between the east direction and the line connecting X and T:
90° - 36° = 54°
Step 3: Calculate the bearing angle between point Y and the tree.
We know that the tree is on bearing S 78° W from point Y. This means that the angle between the line connecting Y and T and the south direction is 78°. However, we want to find the angle between the line connecting Y and T and the east direction.
To find the angle between the east direction and the line connecting Y and T, we subtract the given angle of 78° from 90° (since the east direction is perpendicular to the south direction).
Angle between the east direction and the line connecting Y and T:
90° - 78° = 12°
Step 4: Calculate the distance between points X and Y.
Given that point X is 200m due east of point Y, the distance between X and Y is 200m.
Step 5: Apply the Law of Sines to find the distance between point Y and the tree.
The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of the opposite angle is constant. In this case, we can use it to find the distance between point Y and the tree.
sin(54°) / x = sin(12°) / 200m
To solve for x, isolate it:
x = (sin(54°) / sin(12°)) * 200m
Using a calculator, we can find:
x ≈ 595.48m
Therefore, the distance between point Y and the tree is approximately 595.48m. Rounded to the nearest meter, it would be 595 meters.
To solve this problem, we can use the Law of Cosines.
Let's denote the distance from point x to the tree as a, the distance from point y to the tree as b, and the distance from point x to y as c.
From the given information, we know that x is 200m due east of y, which means that c = 200m.
Now, let's apply the Law of Cosines:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, since angle C is opposite side c, and we want to find side b, we can rearrange the equation to solve for b:
b^2 = a^2 + c^2 - 2ac*cos(C)
Given that angle C is 78° and angle C is the angle between sides a and b, we have:
b^2 = a^2 + 200^2 - 2 * a * 200 * cos(78°)
Now, we can substitute the information that the tree is on bearing s 36° w from point x and s 78° w from point y.
Since the bearing s 36° w is opposite angle A, we have:
A = 180° - 36° = 144°
Since the bearing s 78° w is opposite angle B, we have:
B = 180° - 78° = 102°
Now, we can calculate the value of angle C:
C = 180° - A - B = 180° - 144° - 102° = -66°
However, since the angle cannot be negative, we need to convert it to a positive angle by adding 360°:
C = -66° + 360° = 294°
Now, we can substitute the values into the equation:
b^2 = a^2 + 200^2 - 2 * a * 200 * cos(294°)
To find the distance of the tree from point y, we need to calculate the value of b. We can use trial and error with different values of a to find the value of b that satisfies all the given conditions. Starting with a = 1m:
b^2 = 1^2 + 200^2 - 2 * 1 * 200 * cos(294°)
b^2 = 1 + 40000 - 400 * cos(294°)
b^2 = 1 + 40000 + 400 * cos(66°)
Taking the square root of both sides:
b = sqrt(1 + 40000 + 400 * cos(66°))
Using a calculator, b is approximately equal to 201.13m.
Therefore, the distance of the tree from point y is approximately 201m to the nearest meter.
Draw a diagram. You know two sides and the included angle.
use the law of cosines.