Sure, let's go through each part of the question step by step:
### 1. Exponential Function for Ben's Post
Ben initially shares his post with 2 friends. Each subsequent day, the number of shares triples. The exponential function representing this spread is:
[tex]\[ f(x) = 2(3)^x \][/tex]
This means that the initial number of shares (when [tex]\( x = 0 \)[/tex]) is 2, and it triples every day ([tex]\( 3^x \)[/tex]).
### 2. Exponential Function for Carter's Post
Carter shared his post with 10 friends, and each of those friends shares it with 2 people each day. The exponential growth function for Carter's share can be modeled as:
[tex]\[ f(x) = 10(2)^x \][/tex]
So, Carter's initial shares (when [tex]\( x = 0 \)[/tex]) is 10, and it doubles every day ([tex]\( 2^x \)[/tex]).
### 3. Graphing the Functions
To graph the functions, we'll evaluate each function at [tex]\( x = 0 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = 2 \)[/tex].
For Ben's shares:
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( f(1) = 2 \times 3 = 6 \)[/tex]
- [tex]\( f(2) = 2 \times 3^2 = 18 \)[/tex]
For Carter's shares:
- [tex]\( f(0) = 10 \)[/tex]
- [tex]\( f(1) = 10 \times 2 = 20 \)[/tex]
- [tex]\( f(2) = 10 \times 2^2 = 40 \)[/tex]
For Amber's shares (original function):
- [tex]\( f(0) = 3(4)^0 = 3 \)[/tex]
- [tex]\( f(1) = 3(4)^1 = 12 \)[/tex]
- [tex]\( f(2) = 3(4)^2 = 48 \)[/tex]
For Amber's shares (new function including mailing):
[tex]\[ f(x) = 3(4)^x + 45 \][/tex]
- [tex]\( f(0) = 3(4)^0 + 45 = 48 \)[/tex]
- [tex]\( f(1) = 3(4)^1 + 45 = 57 \)[/tex]
- [tex]\( f(2) = 3(4)^2 + 45 = 93 \)[/tex]
To visualize these functions together, you plot each of the points on a coordinate plane, labeling each curve distinctly.
### 4. Predicting Shares on Day 3 and Day 10
Using the functions, we can predict the number of shares:
For Ben:
- Day 3: [tex]\( f(3) = 2(3)^3 = 54 \)[/tex]
- Day 10: [tex]\( f(10) = 2(3)^{10} = 118098 \)[/tex]
For Carter:*
- Day 3: [tex]\( f(3) = 10(2)^3 = 80 \)[/tex]
- Day 10: [tex]\( f(10) = 10(2)^{10} = 10240 \)[/tex]
### 5. Comparing Amber's Original and New Graph
Amber's original function is [tex]\( 3(4)^x \)[/tex] and her new function, after mailing copies, is [tex]\( 3(4)^x + 45 \)[/tex].
- The original shares at Day 2: [tex]\( f(2) = 3(4)^2 = 48 \)[/tex]
- The new shares at Day 2: [tex]\( f(2) = 3(4)^2 + 45 = 93 \)[/tex]
The new graph will show a vertical shift upwards by 45 units, as evidenced by the addition of the 45 printing. The new graph at any point will consistently be 45 units higher than the original graph.
### 6. Fastest Traveling Post
By comparing the exponential growth rates of the posts:
- Ben: [tex]\( 2(3)^x \)[/tex] (base of 3)
- Carter: [tex]\( 10(2)^x \)[/tex] (base of 2)
- Amber: [tex]\( 3(4)^x \)[/tex] (base of 4)
Amber's post spreads the fastest because it has the highest base (4), showing a more rapid growth rate compared to the others.
### 7. Preference for Post Sharing
Choosing between fewer friends initially but more shares per friend (Amber) or more friends initially but fewer shares:
Given Amber's exponential growth with a base of 4, this results in a faster-growing spread. Despite waking fewer friends initially, the rate of increase (4 per day) outpaces the others over time.
Using our previous calculations:
- By Day 3, Amber already surpasses others with 48 shares compared to Carter's 80 and Ben's 54.
- By Day 10, Amber's shares would skyrocket past the others if calculated, proving exponential growth's substantial impact.
Hence, I would prefer Amber's strategy as it results in significantly higher shares quickly due to faster exponential growth.
