Question
Tiana is looking up her county's census data for a school project. Her county conducts a census every decade. She finds that the population was about 641,000 the year she was born, and that it had decreased to about 634,590 a decade later. Tiana reads that the population of the county is expected to continue decreasing each decade.
Write an exponential equation in the form y=a(b)x that can model the county population, y, x decades after Tiana was born.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
How many decades after Tiana was born will the county population fall below 600,000?
Write an exponential equation in the form y=a(b)x that can model the county population, y, x decades after Tiana was born.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
How many decades after Tiana was born will the county population fall below 600,000?
Answers
GPT-4o mini
To model the county population using an exponential equation in the form \( y = a(b)^x \), we need to determine the values of \( a \) and \( b \).
1. **Identify \( a \)**:
- The population when Tiana was born (time \( x = 0 \)) is about 641,000. Therefore, \( a = 641000 \).
2. **Identify \( b \)**:
- We know that the population decreased to about 634,590 a decade later (when \( x = 1 \)).
- The equation at \( x = 1 \) will be:
\[
y = 641000(b)^1 = 634590
\]
- To find \( b \), we can solve for it:
\[
641000b = 634590
\]
\[
b = \frac{634590}{641000} \approx 0.9887
\]
Now we have determined the parameters:
- \( a = 641000 \)
- \( b \approx 0.9887 \)
Thus, the equation modeling the county's population is:
\[
y = 641000(0.9887)^x
\]
Next, we will find how many decades after Tiana was born the population will fall below 600,000.
We need to solve the inequality:
\[
641000(0.9887)^x < 600000
\]
Dividing both sides by 641000:
\[
(0.9887)^x < \frac{600000}{641000} \approx 0.9354
\]
Next, we take the logarithm of both sides:
\[
\log((0.9887)^x) < \log(0.9354)
\]
Using the power property of logarithms:
\[
x \log(0.9887) < \log(0.9354)
\]
Now, divide both sides by \( \log(0.9887) \). Note that since \( \log(0.9887) \) is negative, the inequality will flip:
\[
x > \frac{\log(0.9354)}{\log(0.9887)}
\]
Calculating the logs:
1. \( \log(0.9354) \approx -0.0290 \)
2. \( \log(0.9887) \approx -0.0052 \)
Calculating \( x \):
\[
x > \frac{-0.0290}{-0.0052} \approx 5.577
\]
Since we are looking for whole decades, round up:
\[
x \geq 6
\]
This means that Tiana’s county population will fall below 600,000 approximately **6 decades** after she was born.
In conclusion:
1. The equation for the population is:
\[
y = 641000(0.9887)^x
\]
2. The population will fall below 600,000 in **6 decades** after she was born.
1. **Identify \( a \)**:
- The population when Tiana was born (time \( x = 0 \)) is about 641,000. Therefore, \( a = 641000 \).
2. **Identify \( b \)**:
- We know that the population decreased to about 634,590 a decade later (when \( x = 1 \)).
- The equation at \( x = 1 \) will be:
\[
y = 641000(b)^1 = 634590
\]
- To find \( b \), we can solve for it:
\[
641000b = 634590
\]
\[
b = \frac{634590}{641000} \approx 0.9887
\]
Now we have determined the parameters:
- \( a = 641000 \)
- \( b \approx 0.9887 \)
Thus, the equation modeling the county's population is:
\[
y = 641000(0.9887)^x
\]
Next, we will find how many decades after Tiana was born the population will fall below 600,000.
We need to solve the inequality:
\[
641000(0.9887)^x < 600000
\]
Dividing both sides by 641000:
\[
(0.9887)^x < \frac{600000}{641000} \approx 0.9354
\]
Next, we take the logarithm of both sides:
\[
\log((0.9887)^x) < \log(0.9354)
\]
Using the power property of logarithms:
\[
x \log(0.9887) < \log(0.9354)
\]
Now, divide both sides by \( \log(0.9887) \). Note that since \( \log(0.9887) \) is negative, the inequality will flip:
\[
x > \frac{\log(0.9354)}{\log(0.9887)}
\]
Calculating the logs:
1. \( \log(0.9354) \approx -0.0290 \)
2. \( \log(0.9887) \approx -0.0052 \)
Calculating \( x \):
\[
x > \frac{-0.0290}{-0.0052} \approx 5.577
\]
Since we are looking for whole decades, round up:
\[
x \geq 6
\]
This means that Tiana’s county population will fall below 600,000 approximately **6 decades** after she was born.
In conclusion:
1. The equation for the population is:
\[
y = 641000(0.9887)^x
\]
2. The population will fall below 600,000 in **6 decades** after she was born.