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Answer :
To determine when the population of the community will reach 23,196, we'll use the given population function:
[tex]\[ P(x) = 18,940 \sqrt{0.05x + 1} \][/tex]
### Step-by-Step Solution:
1. Set the Population Function to the Target Population:
We want to find when the population reaches 23,196:
[tex]\[ 23,196 = 18,940 \sqrt{0.05x + 1} \][/tex]
2. Solve for [tex]\( \sqrt{0.05x + 1} \)[/tex]:
Divide both sides by 18,940 to isolate the square root term:
[tex]\[ \sqrt{0.05x + 1} = \frac{23,196}{18,940} \][/tex]
3. Calculate the Right Side:
Compute the division:
[tex]\[ \sqrt{0.05x + 1} \approx 1.225 \][/tex]
4. Remove the Square Root by Squaring Both Sides:
[tex]\[ 0.05x + 1 = (1.225)^2 \][/tex]
5. Calculate the Square:
[tex]\[(1.225)^2 \approx 1.500625 \][/tex]
6. Isolate the Term with [tex]\( x \)[/tex]:
Subtract 1 from both sides:
[tex]\[ 0.05x = 1.500625 - 1 \][/tex]
[tex]\[ 0.05x = 0.500625 \][/tex]
7. Solve for [tex]\( x \)[/tex]:
Divide both sides by 0.05:
[tex]\[ x = \frac{0.500625}{0.05} \][/tex]
[tex]\[ x \approx 10.0125 \][/tex]
8. Round to the Nearest Whole Number:
Since time is measured in whole years, round:
[tex]\[ x \approx 10 \][/tex]
Therefore, the population will reach 23,196 approximately 10 years from the initial time.
[tex]\[ P(x) = 18,940 \sqrt{0.05x + 1} \][/tex]
### Step-by-Step Solution:
1. Set the Population Function to the Target Population:
We want to find when the population reaches 23,196:
[tex]\[ 23,196 = 18,940 \sqrt{0.05x + 1} \][/tex]
2. Solve for [tex]\( \sqrt{0.05x + 1} \)[/tex]:
Divide both sides by 18,940 to isolate the square root term:
[tex]\[ \sqrt{0.05x + 1} = \frac{23,196}{18,940} \][/tex]
3. Calculate the Right Side:
Compute the division:
[tex]\[ \sqrt{0.05x + 1} \approx 1.225 \][/tex]
4. Remove the Square Root by Squaring Both Sides:
[tex]\[ 0.05x + 1 = (1.225)^2 \][/tex]
5. Calculate the Square:
[tex]\[(1.225)^2 \approx 1.500625 \][/tex]
6. Isolate the Term with [tex]\( x \)[/tex]:
Subtract 1 from both sides:
[tex]\[ 0.05x = 1.500625 - 1 \][/tex]
[tex]\[ 0.05x = 0.500625 \][/tex]
7. Solve for [tex]\( x \)[/tex]:
Divide both sides by 0.05:
[tex]\[ x = \frac{0.500625}{0.05} \][/tex]
[tex]\[ x \approx 10.0125 \][/tex]
8. Round to the Nearest Whole Number:
Since time is measured in whole years, round:
[tex]\[ x \approx 10 \][/tex]
Therefore, the population will reach 23,196 approximately 10 years from the initial time.
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