This document contains examples and explanations of exponential growth, half-life, falling objects, encryption, decryption, and calculating the area of a triangle. It provides sample problems, solutions, and step-by-step workings for each topic. Key concepts include how populations grow exponentially over time, how radioactive materials decay over half-lives, how velocity changes with falling objects, how to encrypt and decrypt messages using matrices, and how to use determinants to calculate the area of a triangle from its vertex coordinates.

1. As we know
Y (0) = y°
Put t = 0 in
Y (0) = Aekt
Y (0) = A
So A= y°
Exponential Growth Example Solution:
we have y0 = 500
t= 3.5
The number of cells doubles every 3.5 hours, so after 3.5 hours there should be 1000
cells.
Therefore after 12 hours there will be cells in the
dish.

2. HALF LIFE:
Example 1 solution

3. Example 2 solution
Falling object example
Hence v(0)=0
So
C=0.
Put again in v
So
V= -9.81t

4. Encryption example
18 5
A =
4 0
18 21
13 0
B =
4 -2
-1 3
67 -21
X = A B =
16 -8
51 27
52 -26

5. The message that you would pass on to the other person is the stream of numbers 67, -21, 16, -8, 51,
27, 52, -26.
DECRYPTION
67 -21
X =
16 -8
51 27
52 -26
B-1
=
0.3 0.2
0.1 0.4
18 5
A = X B-1
=
4 0
18 21
13 0
The receiver then takes the matrix and breaks it apart into values 18, 5, 4, 0, 18, 21, 13, 0 and converts
each of those into a character according to the numbering scheme. 18=R, 5=E, 4=D, 0=space, 18=R,
21=U, 13=M, 0=space.

6. Triangle area
Evaluate that determinant. I'll expand on column 1.
-2 2 1
1 5 1 = + (-2) 5 1 - 1 2 1 + 6 2 1
6 -1 1 -1 1 -1 1 5 1
= -2 ( 5 + 1 ) - 1 ( 2 + 1 ) + 6 ( 2 - 5 ) = -2 ( 6 ) - 1 ( 3 ) + 6 ( -3 ) = -12 - 3 - 18 = -33.
It is possible that you will get a negative determinant, like we did here. Don't worry
about that. The sign is determined by the order you put the points in and can be
easily changed just by switching two rows of the determinant. Area, on the other
hand, can't be negative, so if you get a negative, just drop the sign and make it
positive. Finally, divide it by 2 to find the area.
| -33 | = 33
33 / 2 = 16.5, which was the area.