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Answer :
To solve this problem, we want to find out which inequality describes the scenario of Jen's school tuition increasing by 5% each year while staying below [tex]$20,000 over the next \( n \) years.
Here's how we can figure that out:
1. Understand the problem: The tuition is currently $[/tex]12,000 and grows by 5% each year. We need to find when it will still be less than [tex]$20,000.
2. Exponential growth formula: The tuition increases by a factor of \( 1.05 \) each year. This is because increasing by 5% means you take the full amount (100% or 1.00 of the original amount) and add 5% (0.05), giving you a total of \( 1.00 + 0.05 = 1.05 \).
3. Express the future tuition after \( n \) years: The formula for calculating the future tuition is:
\[
\text{Future Tuition} = 12000 \times (1.05)^n
\]
This equation shows that each year, the tuition is multiplied by 1.05.
4. Set up the inequality: We want the future tuition to be less than $[/tex]20,000:
[tex]\[
12000 \times (1.05)^n < 20000
\][/tex]
5. Identify the correct option: This inequality matches option C: [tex]\( 12000(1.05)^n < 20000 \)[/tex].
So, the correct inequality to model the problem is:
[tex]\[ \text{Option C: } 12000(1.05)^n < 20000 \][/tex]
This inequality will allow you to solve for [tex]\( n \)[/tex], determining in how many years the tuition remains below $20,000, given the annual 5% increase.
Here's how we can figure that out:
1. Understand the problem: The tuition is currently $[/tex]12,000 and grows by 5% each year. We need to find when it will still be less than [tex]$20,000.
2. Exponential growth formula: The tuition increases by a factor of \( 1.05 \) each year. This is because increasing by 5% means you take the full amount (100% or 1.00 of the original amount) and add 5% (0.05), giving you a total of \( 1.00 + 0.05 = 1.05 \).
3. Express the future tuition after \( n \) years: The formula for calculating the future tuition is:
\[
\text{Future Tuition} = 12000 \times (1.05)^n
\]
This equation shows that each year, the tuition is multiplied by 1.05.
4. Set up the inequality: We want the future tuition to be less than $[/tex]20,000:
[tex]\[
12000 \times (1.05)^n < 20000
\][/tex]
5. Identify the correct option: This inequality matches option C: [tex]\( 12000(1.05)^n < 20000 \)[/tex].
So, the correct inequality to model the problem is:
[tex]\[ \text{Option C: } 12000(1.05)^n < 20000 \][/tex]
This inequality will allow you to solve for [tex]\( n \)[/tex], determining in how many years the tuition remains below $20,000, given the annual 5% increase.
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