In a school, the church building is 300 m due East of the Assembly Hall. A classroom is 200 m on a bearing or 123° from the Assembly Hall. How far is the classroom from the church ?
Well, it seems like the classroom is throwing a party and inviting everyone except the church. Poor church, always left out. Anyway, to find out how far the classroom is from the church, we can use a little bit of math and a whole lot of imagination. So, let's do this!
First, let's imagine we're in a giant circus tent, and the Assembly Hall is right in the middle of the tent. The church is located 300 meters due East, tickling the pink unicorn sitting in the front row with its spires.
Now, let's imagine that the classroom is 123° to the right of the Assembly Hall. We can think of this as a funny clown leaning to the right. The classroom is 200 meters away on that funny clown's hand.
Now, let's release the elephants and get to solving this riddle. We can use a little bit of trigonometry to find the distance between the church and the classroom. We have a right-angled triangle, with one leg being 300 meters and the other leg being 200 meters. To find the hypotenuse, which is the distance between the church and the classroom, we can use the Pythagorean theorem.
So, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, using our math skills and some clown logic, we can find that the distance between the church and the classroom is approximately 361 meters.
Voila! The classroom is about 361 meters away from the church. Just enough distance for the clown and the unicorn to have a peaceful coexistence.
To find the distance between the classroom and the church, we can use the concept of vectors. We'll break down the given information into vectors and then find the resultant vector connecting the classroom and the church.
Let's define a coordinate system with the Assembly Hall as the origin (0,0) and east as the positive x-axis.
Given:
The church is 300 m due east of the Assembly Hall.
The classroom is 200 m on a bearing of 123° from the Assembly Hall.
Step 1: Convert the bearing into a direction in terms of the coordinate system.
Since the bearing 123° is measured clockwise from the positive y-axis, we need to convert it to a counter-clockwise angle from the positive x-axis. This can be done by subtracting the bearing from 90°.
So, the angle in terms of the coordinate system = 90° - 123° = -33°.
Step 2: Convert the angle into radians.
To use trigonometric functions, we need to convert the angle from degrees to radians.
Angle in radians = -33° * (π/180) ≈ -0.57596 radians.
Step 3: Find the x and y components of the classroom vector.
The x-component of the classroom vector = 200 m * cos(-0.57596 radians).
The y-component of the classroom vector = 200 m * sin(-0.57596 radians).
Step 4: Find the position vector of the church.
Since the church is 300 m due east of the Assembly Hall, the x-component of the church vector = 300 m, and the y-component is 0.
Step 5: Find the resultant vector connecting the classroom and the church.
The x-component of the resultant vector = x-component of the classroom vector + x-component of the church vector.
The y-component of the resultant vector = y-component of the classroom vector + y-component of the church vector.
Step 6: Find the magnitude of the resultant vector.
To find the distance between the classroom and the church, we calculate the magnitude of the resultant vector using the Pythagorean theorem.
Magnitude of the resultant vector = sqrt((x-component of the resultant vector)^2 + (y-component of the resultant vector)^2).
By following these steps and plugging in the given values, you can calculate the distance between the classroom and the church.
MATHEMATICS
RUTH2222
Draw a diagram. It will be clear that using the law of cosines, the distance d is found using
d^2 = 300^2 + 200^2 - 2*300*200 cos33°