To determine the value of the index at the end of the day, follow these steps:
1. Calculate the initial value of the index:
- For Stock [tex]\(X\)[/tex]: The value is [tex]\(5000 \text{ shares} \times \$4.30 \text{ per share} = \$21500\)[/tex].
- For Stock [tex]\(Y\)[/tex]: The value is [tex]\(2000 \text{ shares} \times \$3.20 \text{ per share} = \$6400\)[/tex].
- For Stock [tex]\(Z\)[/tex]: The value is [tex]\(8000 \text{ shares} \times \$4.90 \text{ per share} = \$39200\)[/tex].
- Summing these values gives the total initial value of the index:
[tex]\[
\$21500 + \$6400 + \$39200 = \$67100
\][/tex]
2. Determine the percentage increase over the day:
- The index rises by [tex]\(4.9\%\)[/tex].
- The new value of the index at the end of the day is calculated as follows:
[tex]\[
\text{New total value} = \text{Initial total value} \times (1 + \text{Percentage increase})
\][/tex]
- Substituting the given values:
[tex]\[
\text{New total value} = \$67100 \times (1 + 0.049) = \$67100 \times 1.049 = \$70387.9
\][/tex]
3. Round the final value to the nearest hundred:
- The value \[tex]$70387.9 rounds to \$[/tex]70400.
Thus, the value of the index at the end of the day, rounded to the nearest hundred, is:
[tex]\[
\boxed{\$70400}
\][/tex]
