To find the diver's depth in relation to the surface of the water, let's break down the problem step-by-step:
1. Convert the mixed numbers to improper fractions:
- The initial dive is [tex]\(17 \frac{2}{3}\)[/tex] yards.
[tex]\[
17 \frac{2}{3} = 17 + \frac{2}{3} = \frac{17 \times 3 + 2}{3} = \frac{51 + 2}{3} = \frac{53}{3} \text{ yards}
\][/tex]
- The deeper dive is [tex]\(9 \frac{3}{8}\)[/tex] yards.
[tex]\[
9 \frac{3}{8} = 9 + \frac{3}{8} = \frac{9 \times 8 + 3}{8} = \frac{72 + 3}{8} = \frac{75}{8} \text{ yards}
\][/tex]
2. Sum the depths to find the total depth:
- The total depth is the sum of the two improper fractions. To sum these, we need a common denominator. The least common multiple (LCM) of the denominators 3 and 8 is 24.
- Convert each fraction to have the common denominator:
[tex]\[
\frac{53}{3} = \frac{53 \times 8}{3 \times 8} = \frac{424}{24}
\][/tex]
[tex]\[
\frac{75}{8} = \frac{75 \times 3}{8 \times 3} = \frac{225}{24}
\][/tex]
- Add the two fractions:
[tex]\[
\frac{424}{24} + \frac{225}{24} = \frac{424 + 225}{24} = \frac{649}{24}
\][/tex]
3. Convert the improper fraction to a mixed number:
- Divide 649 by 24 to get the mixed number.
[tex]\[
649 \div 24 \approx 27.041666666666668
\][/tex]
- This means:
[tex]\[
649 = 27 \times 24 + 1 \quad (\text{as } 27 \times 24 = 648, \text{ with a remainder of 1})
\][/tex]
[tex]\[
\text{Thus, } \frac{649}{24} = 27 \frac{1}{24}
\][/tex]
4. Determine the diver's depth in relation to the surface:
- Since the diver is below the surface, the total depth will be negative.
[tex]\[
-27 \frac{1}{24}
\][/tex]
Therefore, the diver's depth in relation to the surface of the water is:
[tex]\[
-27 \frac{1}{24} \text{ yards}
\][/tex]
This is the correct depth in relation to the surface of the water. Thus, the correct answer is:
[tex]\[
\boxed{-27 \frac{1}{24}}
\][/tex]
