QUESTION
Harold and Kumar drove their car 10 kilometers north from their house to the theatres. After they finished their movie, they headed 5 kilometers west, to White Castle to get some food. How far are they from their house? In which direction does Harold and Kumar have to travel to get home and in the shortest possible route?
ANSWER
First of all I would draw a diagram that looks like so to iillustrate what this question is asking:
So from here, we are given two sides. One going north, and the other going west, which gives us right angle. To figure out how far they are from White Castle to their house we must use the Pythagorean Theory. We all know the Pythagorean theory right? Well you should know it, we learned this a while back. We can only use this theory if it is a right angle triangle. The Pythagorean Theory formula is:
C represents the hypotenuse ( the side opposite the right angle) and A and B are the lengths of the two other sides.
For this question, we are missing the hypotenuse ( the distance from White Castle to their house). To figure out the hypotenuse we must plug in the numbers of our two given sides into either A or B. We do this to figure out our missing side, which is side C. Once we plug in the numbers into the formula, it should look like this :
We then solve for C, the steps are as follows:
We now have figured out how far they are from White Castle to their house. Which is 11.18 km away.We now got to figure out which direction they need to go to get home from White Castle. There are 4 different notations to indicate the direction they need to go.
-An Arrow
-Bearing
-[direction] [angle] [direction]
-[angle] [direction] of [direction]
Here is all the information we have figured out:
We can easily answer the direction they need to go with an arrow, which is South East. But the next 3 notations are a little harder because we got to figure out our angles.
To figure out an angle we could use the “Sine Law” which is:
The only angle we got is angle A, which is 90 degrees. To figure out the rest of the angles we need to use the sine law. Using the labeled diagram above we plug in the number to figure out the angles:
This is how our diagram looks like now with everything put together:
So the shortest possible way home from White Castle is:
As a bearing : Bearing [ 153.44 degrees]
How to find a bearing you ask? Well first you start at North then go down to your desired angle.
Here our desired angle was 63.44 degrees South of West, but were missing our 90 degree angle from from North to East. We then add these two angles to figure out the bearing:
90 + 63.44 = 153.44 degrees
The next two should be straight forward. You would state the two directions your angle makes and state the degree of the angle.
[direction] [angle] [direction]: West 63.44 degrees South
[angle] [ direction] of [direction] : 63.44 degrees South of West
My second problem is related to the unit "Matrices."
Write a matrix to summarize the networks.
First of all I would make a matrix showing the directions of the networks. 1 meaning there's a route going to that point and 0 meaning there is no route to that point. Here is the matrix diagram:
So we answered the first question. Now can we multiply matrices? Well of course. You can only multiply matrices if the number of rows on your first matrix is equal to the number of columns of the second matrix. Like the diagram shown below which are both 2 x2 matrices:
To multiply matrices we got to figure out the digits of the row in our first matrix and the digits of the column of the second matrix. It's weird how we multiply matrices so a diagram should help a little better :
So now we multiply 3C with 1A then add 3D and 2A which gives us:
1(2) + 7 (4 ) = 30
Now we sort of get what we're being asked to do. So, to continue multiplying the matrix here is how we do the rest of the multiplication:
So this is how we multiply everything out:
3C(1A) + 3D(2A) =
3C(1B) + 3D(2B) =
4C(1A) + 4D(2A) =
4C(1B) + 4D(2B) =
Multiplying with the digits:
1(2) + 7(4) = 30
1(8) + 7(6) = 50
3(2) + 5(4) = 26
3(8) + 5(6) =54
So now our resultant matrix looks like this :
