To solve the problem of determining how many square feet of Marcus's wall will be painted blue, we need to go through a few steps systematically:
1. Determine the wall's height and length in mixed numbers and then convert them to improper fractions or decimals for ease of calculation.
- The height of the wall is [tex]\(8 \frac{2}{5}\)[/tex] feet. This can be converted to a decimal as follows:
[tex]\[
8 \frac{2}{5} = 8 + \frac{2}{5} = 8 + 0.4 = 8.4 \text{ feet}
\][/tex]
- The length of the wall is [tex]\(16 \frac{2}{3}\)[/tex] feet. This can be converted to a decimal as follows:
[tex]\[
16 \frac{2}{3} = 16 + \frac{2}{3} = 16 + 0.6667 = 16.6667 \text{ feet}
\][/tex]
2. Calculate the total area of the wall by multiplying the height by the length.
Using the decimal values obtained:
[tex]\[
\text{Total area} = \text{height} \times \text{length} = 8.4 \times 16.6667
\][/tex]
Performing the multiplication:
[tex]\[
8.4 \times 16.6667 = 140.00000000000003 \text{ square feet} \approx 140 \text{ square feet}
\][/tex]
3. Determine the portion of the wall that will be painted blue.
Marcus paints [tex]\(\frac{1}{2}\)[/tex] of the wall blue. Therefore, we need to find [tex]\(\frac{1}{2}\)[/tex] of the total area:
[tex]\[
\text{Blue area} = \frac{1}{2} \times \text{Total area} = \frac{1}{2} \times 140
\][/tex]
Simplifying:
[tex]\[
\text{Blue area} = 70.00000000000001 \approx 70 \text{ square feet}
\][/tex]
So, the number of square feet of the wall that will be painted blue is [tex]\( \boxed{70} \)[/tex].
