Let's solve the problem step by step:
1. First, we need to convert the mixed numbers to improper fractions.
Derek's initial mix amount is:
[tex]\[
2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3}
\][/tex]
The multiplier for the job estimate is:
[tex]\[
2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\][/tex]
2. Multiply these fractions to find the total amount of concrete needed:
[tex]\[
\text{Total amount needed} = \frac{7}{3} \times \frac{5}{2} = \frac{35}{6}
\][/tex]
3. Convert this amount back to a mixed number:
[tex]\[
\frac{35}{6} = 5 \frac{5}{6}
\][/tex]
4. Now, we need to find out how many more cubic yards of concrete are needed beyond the initial mix:
[tex]\[
\text{More needed} = \frac{35}{6} - \frac{7}{3}
\][/tex]
To subtract the fractions, convert [tex]\( \frac{7}{3} \)[/tex] to a fraction with the same denominator as [tex]\( \frac{35}{6} \)[/tex]:
[tex]\[
\frac{7}{3} = \frac{14}{6}
\][/tex]
Then:
[tex]\[
\text{More needed} = \frac{35}{6} - \frac{14}{6} = \frac{21}{6} = 3 \frac{1}{2}
\][/tex]
Therefore, the amount of concrete Derek needs beyond his initial mix is:
[tex]\[
\boxed{3 \frac{1}{2}}
\][/tex]
So, the correct option is B. [tex]\(3 \frac{1}{2}\)[/tex].
