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Here are a few recommended readings before getting started with this lesson.
During the holidays, Ignacio's family plans to visit the theme park Mondo Marino in California. While looking at the park's website to buy tickets, Ignacio found information about a brand new aquarium inside the park.
Wow! Ignacio notices that the park has a great contest. The person who solves a series of problems about the fish population in the aquarium will win an annual family fun pass. The website provides a function that models the fish population's growth.$ Be Growing,so he opens a bank account. He deposits an initial amount of $850 and plans to deposit the same quantity every month. The account earns no interest.
wild, increases by 2.5% every year.
$ Be Growingbeing played non-stop on the local radio.
Marky and his friend Tifanniqua are playing a video game in which they each have to build a city. The game gives them 15000 v-coins for buying supplies. Since this virtual money is not enough, they must invest this initial amount. After some exploration, they have discovered two banks in the game that can help them increase their v-coins.
Considering that one year in the game is about 12 minutes in the real world, the following are the offers of each virtual bank in the game.
Bank B: B(m)=15000(1+120.025)m
Investment Period | Bank A | Bank B |
---|---|---|
A(t)=15000+450t | B(m)=15000(1+120.025)m | |
1 Year (12 Months) | 15450 | ≈15379 |
2 Years | 15900 | ≈15768 |
3 Years | 16350 | ≈16167 |
4 Years | 16800 | ≈16576 |
5 Years | 17250 | ≈16995 |
6 Years | 17700 | ≈17425 |
7 Years | 18150 | ≈17865 |
8 Years | 18600 | ≈18317 |
9 Years | 19050 | ≈18780 |
10 Years | 19500 | ≈19255 |
11 Years | 19950 | ≈19742 |
12 Years | 20400 | ≈20241 |
13 Years | 20850 | ≈20753 |
14 Years | 21300 | ≈21278 |
15 Years | 21750 | ≈21816 |
16 Years | 22200 | ≈22368 |
17 Years | 22650 | ≈22934 |
18 Years | 23100 | ≈23514 |
19 Years | 23550 | ≈24108 |
20 Years | 24000 | ≈24718 |
Interpretation: Bank B generates higher income after the fourteenth year. However, for the previous years, Bank B is more profitable.
It has been obtained that the balance of Marky's investment after 10 years will be 19500 v-coins and Tiffaniqua's balance will be about 19255. Therefore, the offer of Bank A is better for the period of 10 years.
Investment Period in Years | Bank A | Bank B |
---|---|---|
A(t)=15000+450t | B(m)=15000(1+120.025)m | |
1 | 15450 | ≈15379 |
2 | 15900 | ≈15768 |
3 | 16350 | ≈16167 |
4 | 16800 | ≈16576 |
5 | 17250 | ≈16995 |
6 | 17700 | ≈17425 |
7 | 18150 | ≈17865 |
8 | 18600 | ≈18317 |
9 | 19050 | ≈18780 |
10 | 19500 | ≈19255 |
11 | 19950 | ≈19742 |
12 | 20400 | ≈20241 |
13 | 20850 | ≈20753 |
14 | 21300 | ≈21278 |
15 | 21750 | ≈21816 |
16 | 22200 | ≈22368 |
17 | 22650 | ≈22934 |
18 | 23100 | ≈23514 |
19 | 23550 | ≈24108 |
20 | 24000 | ≈24718 |
It can be seen that Bank B generates a higher income after the fourteenth year. However, for the previous years, Bank A is more profitable. The friends have been having such a great time playing video games. In the end, all was just fake money.
Besides singing, Marky is really into fishing. A small company in town called SeaBase Fish specializes in shrimp farming. They currently only farm shrimp, but they want to expand and introduce two types of fish next year, Catfish and Tilapia. The good news for Marky is that they will allow local kids the chance to fish recreationally.
Marky has never caught either type of fish in his life. He is interested in finding information about both types and then sharing this information with his friends. Online, Marky found information from another study about the growth model of each species' population after x weeks of the initial population being introduced into the sea.x | 55x+10 | yC=55x+10 | (x,yC) |
---|---|---|---|
0 | 55(0)+10 | 10 | (0,10) |
2 | 55(2)+10 | 120 | (2,120) |
4 | 55(4)+10 | 230 | (4,230) |
6 | 55(6)+10 | 340 | (6,340) |
8 | 55(8)+10 | 450 | (8,450) |
The data plots can be plotted and connected in a coordinate plane.
Similarly, a table for Tilapia's model will be created.
x | 4⋅2x | yT=4⋅2x | (x,yT) |
---|---|---|---|
0 | 4⋅20 | 4 | (0,4) |
2 | 4⋅22 | 16 | (2,16) |
4 | 4⋅24 | 64 | (4,64) |
6 | 4⋅26 | 256 | (6,256) |
8 | 4⋅28 | 1024 | (8,1024) |
Now, the graph for Tilapia fish can be added by plotting and connecting the data points obtained in the second table.
It can be seen that the functions intersect at x≈6.5. Therefore, it is expected that after about six and a half weeks, the sizes of both populations will be approximately equal. This can be verified by evaluating each function when x=6.5.
Catfish | Tilapia | |
---|---|---|
Growth Function | yC=55x+10 | yT=4⋅2x |
Growth After 621 Weeks | yC=55(6.5)+10 | yT=4⋅26.5 |
Evaluate and Approximate | yC≈368 | yT≈362 |
It can be noted although x=6.5 is not the exact answer, population sizes are very close.
A possible interpretation that Marky can make from his findings is that after six-and-a-half weeks after the fish are introduced, he can focus on catching Tilapia. That interpretation can be made assuming that Marky wants to catch the fish species that has such a high population that it is easier to catch.
Ignacio's family is going to have some large expenses for a holiday trip, so Ignacio's mom asked him for a hand. She wants to post some ads on social media to reach more clients to generate sales for her a client. She represents an up-and-coming musician named Marky. There is one problem, Ignacio has no idea which social media site is the best option.
For this reason, Ignacio collected some data about how the number of users of these social media sites increases on a weekly basis.
User Growth per Week on Each Social Media (in Thousands) | ||
---|---|---|
Time (in Weeks) | Option 1 | Option 2 |
0 | 18 | 15 |
1 | 21 | 18 |
2 | 24 | 21.5 |
3 | 27 | 26 |
4 | 30 | 31 |
5 | 33 | 37.5 |
Ignacio wonders if this data can help him make the best decision. Find the following information to help him decide where they should post the ads.
Option 2: Exponential
Option 2: g(x)=15⋅1.2x
Option 2: ≈133700 users
The number of users of Option 1 increases by a constant of 3 thousands every week. This data follows a linear model. Using the same process, the pattern for Option 2 will be found.
Each week, the number of users of Option 2 grows by a factor of about 1.2. That means this data follows an exponential model.
x=0, f(x)=18
Zero Property of Multiplication
Rearrange equation
x=0, g(x)=15
a0=1
Identity Property of Multiplication
Rearrange equation
x=12
Calculate power
Multiply
Sells are doing great. Ignacio's family can now have an even more fantastic holiday at the theme park.
x=0
a0=1
Identity Property of Multiplication
f(x)=850, x=25
LHS/100=RHS/100
Calculate quotient
LHS251=RHS251
(am)n=am⋅n
25⋅25a=a
a1=a
Rearrange equation
Use a calculator
Round to 2 decimal place(s)