Answer :
To find the inequality that describes when the expected value of card [tex]\( A \)[/tex] is at least the predicted value of card [tex]\( B \)[/tex], we need to compare their future values after [tex]\( t \)[/tex] years.
### Step-by-Step Solution:
1. Identify the Current Values and Growth Rates:
- Current value of baseball card [tex]\( A \)[/tex] is \[tex]$50. - The value of card \( A \) increases exponentially by a factor of \( 1.06 \) every 2 years. - Current value of baseball card \( B \) is \$[/tex]40.
- The value of card [tex]\( B \)[/tex] increases exponentially by a factor of [tex]\( 1.08 \)[/tex] every 2 years.
2. Express the Future Values:
- The future value of card [tex]\( A \)[/tex] after [tex]\( t \)[/tex] years will multiply by [tex]\( 1.06 \)[/tex] every 2 years. Hence, after [tex]\( t \)[/tex] years, the value of card [tex]\( A \)[/tex] will be:
[tex]\[ 50 \times (1.06)^{\frac{t}{2}} \][/tex]
- Similarly, the future value of card [tex]\( B \)[/tex] after [tex]\( t \)[/tex] years will multiply by [tex]\( 1.08 \)[/tex] every 2 years. Hence, after [tex]\( t \)[/tex] years, the value of card [tex]\( B \)[/tex] will be:
[tex]\[ 40 \times (1.08)^{\frac{t}{2}} \][/tex]
3. Formuate the Inequality:
To find the time [tex]\( t \)[/tex] when the value of card [tex]\( A \)[/tex] is at least the value of card [tex]\( B \)[/tex], we need the future value of card [tex]\( A \)[/tex] to be greater than or equal to the future value of card [tex]\( B \)[/tex]. Thus, we set up the inequality:
[tex]\[ 50 \times (1.06)^{\frac{t}{2}} \geq 40 \times (1.08)^{\frac{t}{2}} \][/tex]
4. Simplify the Exponents:
Since the exponential factors are for every 2 years, we rephrase the inequality in terms of [tex]\( 2t \)[/tex]:
[tex]\[ 50 \times (1.06)^{2t} \geq 40 \times (1.08)^{2t} \][/tex]
5. Identify the Correct Option:
Therefore, the inequality can be written as:
[tex]\[ 50(1.06)^{2t} \geq 40(1.08)^{2t} \][/tex]
After comparing this to the provided options, we see that the correct inequality is found in option [tex]\( A \)[/tex]:
[tex]\[ 50(1.06)^{2t} \geq 40(1.08)^{2t} \][/tex]
Thus, the correct inequality and answer is:
Option A: [tex]\( 50(1.06)^{2t} \geq 40(1.08)^{2t} \)[/tex].
### Step-by-Step Solution:
1. Identify the Current Values and Growth Rates:
- Current value of baseball card [tex]\( A \)[/tex] is \[tex]$50. - The value of card \( A \) increases exponentially by a factor of \( 1.06 \) every 2 years. - Current value of baseball card \( B \) is \$[/tex]40.
- The value of card [tex]\( B \)[/tex] increases exponentially by a factor of [tex]\( 1.08 \)[/tex] every 2 years.
2. Express the Future Values:
- The future value of card [tex]\( A \)[/tex] after [tex]\( t \)[/tex] years will multiply by [tex]\( 1.06 \)[/tex] every 2 years. Hence, after [tex]\( t \)[/tex] years, the value of card [tex]\( A \)[/tex] will be:
[tex]\[ 50 \times (1.06)^{\frac{t}{2}} \][/tex]
- Similarly, the future value of card [tex]\( B \)[/tex] after [tex]\( t \)[/tex] years will multiply by [tex]\( 1.08 \)[/tex] every 2 years. Hence, after [tex]\( t \)[/tex] years, the value of card [tex]\( B \)[/tex] will be:
[tex]\[ 40 \times (1.08)^{\frac{t}{2}} \][/tex]
3. Formuate the Inequality:
To find the time [tex]\( t \)[/tex] when the value of card [tex]\( A \)[/tex] is at least the value of card [tex]\( B \)[/tex], we need the future value of card [tex]\( A \)[/tex] to be greater than or equal to the future value of card [tex]\( B \)[/tex]. Thus, we set up the inequality:
[tex]\[ 50 \times (1.06)^{\frac{t}{2}} \geq 40 \times (1.08)^{\frac{t}{2}} \][/tex]
4. Simplify the Exponents:
Since the exponential factors are for every 2 years, we rephrase the inequality in terms of [tex]\( 2t \)[/tex]:
[tex]\[ 50 \times (1.06)^{2t} \geq 40 \times (1.08)^{2t} \][/tex]
5. Identify the Correct Option:
Therefore, the inequality can be written as:
[tex]\[ 50(1.06)^{2t} \geq 40(1.08)^{2t} \][/tex]
After comparing this to the provided options, we see that the correct inequality is found in option [tex]\( A \)[/tex]:
[tex]\[ 50(1.06)^{2t} \geq 40(1.08)^{2t} \][/tex]
Thus, the correct inequality and answer is:
Option A: [tex]\( 50(1.06)^{2t} \geq 40(1.08)^{2t} \)[/tex].
