Question
Attendance at AI (Artificial Intelligence)
conferences has increased by about { 26% } every year. The first Al conference took place in 2010 and 9421 people attended an Al conference tha year.
a. Estimate the number of people who attended an AI conference in 2011.
b. Estimate the number of people who attended an Al conference in 2020.
c. Write an equation for P (t), the number of people who attended an Al conference t years after 2010.
conferences has increased by about { 26% } every year. The first Al conference took place in 2010 and 9421 people attended an Al conference tha year.
a. Estimate the number of people who attended an AI conference in 2011.
b. Estimate the number of people who attended an Al conference in 2020.
c. Write an equation for P (t), the number of people who attended an Al conference t years after 2010.
Answers
GPT-4o mini
To solve the problem, we can use the formula for exponential growth. Given that the attendance increases by 26% each year, we can represent this mathematically.
The formula for exponential growth is:
\[ P(t) = P_0 (1 + r)^t \]
where:
- \( P(t) \) is the population (in this case, attendance) at time \( t \),
- \( P_0 \) is the initial amount (initial attendance in this case),
- \( r \) is the growth rate (as a decimal),
- \( t \) is the number of years after the initial year.
### Given Data:
- \( P_0 = 9421 \) (attendance in 2010),
- \( r = 0.26 \) (the growth rate of 26%).
### a. Estimate the number of people who attended an AI conference in 2011.
\[ t = 1 \] (since 2011 is one year after 2010)
Using the formula:
\[
P(1) = 9421 \times (1 + 0.26)^1
\]
\[
P(1) = 9421 \times 1.26 \approx 11803.46
\]
Rounding to the nearest whole number, approximately **11,803 people** attended in 2011.
### b. Estimate the number of people who attended an AI conference in 2020.
\[ t = 10 \] (since 2020 is ten years after 2010)
Using the formula:
\[
P(10) = 9421 \times (1 + 0.26)^{10}
\]
Calculating \( (1 + 0.26)^{10} \):
\[
1.26^{10} \approx 10.781
\]
Now substituting back:
\[
P(10) \approx 9421 \times 10.781 \approx 101,676.7
\]
Rounding to the nearest whole number, approximately **101,677 people** attended in 2020.
### c. Write an equation for \( P(t) \), the number of people who attended an AI conference \( t \) years after 2010.
Using the earlier formula, we substitute:
- \( P_0 = 9421 \)
- \( r = 0.26 \)
The equation is:
\[
P(t) = 9421 \times (1.26)^t
\]
### Summary:
a. Approximately 11,803 attended in 2011.
b. Approximately 101,677 attended in 2020.
c. The equation for attendance is \( P(t) = 9421 \times (1.26)^t \).
The formula for exponential growth is:
\[ P(t) = P_0 (1 + r)^t \]
where:
- \( P(t) \) is the population (in this case, attendance) at time \( t \),
- \( P_0 \) is the initial amount (initial attendance in this case),
- \( r \) is the growth rate (as a decimal),
- \( t \) is the number of years after the initial year.
### Given Data:
- \( P_0 = 9421 \) (attendance in 2010),
- \( r = 0.26 \) (the growth rate of 26%).
### a. Estimate the number of people who attended an AI conference in 2011.
\[ t = 1 \] (since 2011 is one year after 2010)
Using the formula:
\[
P(1) = 9421 \times (1 + 0.26)^1
\]
\[
P(1) = 9421 \times 1.26 \approx 11803.46
\]
Rounding to the nearest whole number, approximately **11,803 people** attended in 2011.
### b. Estimate the number of people who attended an AI conference in 2020.
\[ t = 10 \] (since 2020 is ten years after 2010)
Using the formula:
\[
P(10) = 9421 \times (1 + 0.26)^{10}
\]
Calculating \( (1 + 0.26)^{10} \):
\[
1.26^{10} \approx 10.781
\]
Now substituting back:
\[
P(10) \approx 9421 \times 10.781 \approx 101,676.7
\]
Rounding to the nearest whole number, approximately **101,677 people** attended in 2020.
### c. Write an equation for \( P(t) \), the number of people who attended an AI conference \( t \) years after 2010.
Using the earlier formula, we substitute:
- \( P_0 = 9421 \)
- \( r = 0.26 \)
The equation is:
\[
P(t) = 9421 \times (1.26)^t
\]
### Summary:
a. Approximately 11,803 attended in 2011.
b. Approximately 101,677 attended in 2020.
c. The equation for attendance is \( P(t) = 9421 \times (1.26)^t \).