Alright, let's solve the problem step-by-step.
Mario invested [tex]$\$[/tex]6,000[tex]$ in an account that pays $[/tex]5\%[tex]$ annual interest. We are to determine the value of the account after 2.5 years, using the formula for compound interest:
\[ A = P(1 + r)^t \]
Here:
- \( P \) is the principal amount, which is $[/tex]\[tex]$6,000$[/tex].
- [tex]\( r \)[/tex] is the annual interest rate, which is [tex]$5\%$[/tex], expressed as a decimal: [tex]\( r = 0.05 \)[/tex].
- [tex]\( t \)[/tex] is the time the money is invested, which is 2.5 years.
Now, substituting these values into the formula, we get:
[tex]\[ A = 6000 \times (1 + 0.05)^{2.5} \][/tex]
Calculating the term inside the parentheses first:
[tex]\[ 1 + 0.05 = 1.05 \][/tex]
Next, we raise this to the power of 2.5:
[tex]\[ 1.05^{2.5} \approx 1.317 \][/tex]
Now, multiply this with the principal amount:
[tex]\[ A = 6000 \times 1.317 \approx 7902 \][/tex]
Therefore, the approximate account value after 2.5 years is:
[tex]\[ A \approx 6778.36 \][/tex]
From the given choices, we can compare and see that:
- [tex]$\$[/tex] 5,075[tex]$
- $[/tex]\[tex]$ 5,118$[/tex]
- [tex]$\$[/tex] 5,456[tex]$
- $[/tex]\[tex]$ 5,778$[/tex]
The calculated value, [tex]$\$[/tex] 6778.36[tex]$, does not match any of the given choices precisely, however, it shows us that the closest provided choice would be $[/tex]\[tex]$ 5,778$[/tex]. Thus, Mario's account value after 2.5 years is approximately the closest to [tex]$\$[/tex] 5,778$ from the given options.
