No.39

The altitude, equal sides, and nonequal sides of an isosceles triangle have lengths that are, in the order listed, consecutive even numbers of centimeters.
What is the area of the triangle?

The (even) altitude = 2k for some integer k.
The slanted side = 2k + 2
The base is 2k + 4, so half of it is k + 2.

The altitude of an isosceles triangle splits it into two right triangles,
so from the Pythagorean Theorem we have:
(k + 2)^2 + (2k)^2 = (2k + 2)^2

k^2 + 4k + 4 + 4k^2 = 4k^2 + 8k + 4
k^2 = 4k

k = 0 or 4

0 would not be a "triangle", but if you allow for
the degenerate case of 0 altitude, 2 on the sides and 4 on the base, the numbers work,
and the area is 0.

For k = 4, we have:
altitude 8, sides 10, base 12 (two 6-8-10 right triangles, which also have consecutive even sides)
and the area is 8 * 12/2 = 48

No.38

Adding Vectors

And here is how to add two vectors after breaking them into x and y parts:

The vector (8,13) and the vector (26,7) add up to the vector (34,20)

Example: add the vectors a = (8,13) and b = (26,7)

c = a + b

c = (8,13) + (26,7) = (8+26,13+7) = (34,20)

Subtracting Vectors

Remember: to subtract, first reverse the vector you want to subtract, then add.

Example: subtract k = (4,5) from v = (12,2)

a = v + -k

a = (12,2) + -(4,5) = (12,2) + (-4,-5) = (12-4,2-5) = (8,-3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

|a|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||a||

You can use Pythagoras' theorem to calculate it:

|a| = √( x² + y² )

A vector with magnitude 1 is called a Unit Vector.

Vector vs Scalar

When using vectors we call an ordinary number a "scalar".

Scalar: just a number (like 7 or -0.32) ... definitely not a vector.

A vector is often written in bold,

so c is a vector, and it has magnitude and direction

but c is just a value, like 3 or 12.4

Example: kb is actually the scalar k times the vector b.

No.37

Vectors

This is a vector:

A vector has magnitude (how long it is) and direction:

The length of the line shows its magnitude and the arrowhead points in the direction.

You can add two vectors by simply joining them head-to-tail:

And it doesn't matter which order you add them, you get the same result:

3-D graph

No.35

Graphing polar equations
Limacons 

, where a and b are NOT equal to 0

• If  the graph of the Limacon has an inner loop!
• If  the graph of the Limacon is "dimpled."
• If the graph of the Limacon is considered "convex."

The graphs of Limacons are generated as the angle increases from 0 to 2.

a.   Limacon with inner loop: 
Note that a = 2 and b = 3 and 

b.   Dimpled Limacon: 

Note that a = 3 and b = 2 and 

c.   Convex Limacon:  .   Note that the graph is not quite circular!

Note that a = 8 and b = 2 and 

No.34

The polar coordinate system is a two-dimensional coordinate system in which each
point P on a plane is determined by a distance r from a θ point O that is called the pole (or origin) and an angle from a θ direction. The point P is represented by the ordered pair (r; θ ) and r; θ are called polar coordinates.

No.33

Present Value

So $1,000 now is the same as $1,100 next year (at 10% interest).

We say the Present Value of $1,100 next year is $1,000

Because we could turn $1,000 into $1,100 (if we could earn 10% interest).

Now let us extend this idea further into the future ...

How to Calculate Future Payments

Let us stay with 10% Interest. That means that money grows by 10% every year, like this:

So:

• $1,100 next year is the same as $1,000 now.
• And $1,210 in 2 years is the same as $1,000 now.
• etc

In fact all those amounts are the same (considering when they occur and the 10% interest).