Let us analyze the given problem step by step and derive the exact solution.
### Given:
- Choo Kheng must travel at least 288 kilometers in total.
- The submarine travels on the water's surface for [tex]\(2 \frac{2}{3}\)[/tex] hours.
- The inequality for the travel distance is [tex]\(36S + 64U \geq 288\)[/tex].
### Step 1: Convert the Mixed Number
First, we convert the mixed number [tex]\(2 \frac{2}{3}\)[/tex] into an improper fraction:
[tex]\[ 2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \][/tex]
### Step 2: Express S in Decimal Form
For ease of calculations, we express [tex]\(2 \frac{2}{3}\)[/tex] as a decimal:
[tex]\[ 2 \frac{2}{3} = 2 + \frac{2}{3} = 2 + 0.6666666666666667 = 2.6666666666666665 \][/tex]
### Step 3: Calculate the Surface Distance
The distance traveled on the water's surface, [tex]\(D_s\)[/tex], can be found by plugging [tex]\(S = 2.6666666666666665\)[/tex] into the expression for the surface distance ([tex]\(36S\)[/tex]):
[tex]\[ D_s = 36 \times S = 36 \times 2.6666666666666665 = 96 \text{ kilometers} \][/tex]
### Step 4: Use the Inequality to Solve for U
Now, using the inequality [tex]\(36S + 64U \geq 288\)[/tex]:
[tex]\[ 36 \times 2.6666666666666665 + 64U \geq 288 \][/tex]
[tex]\[ 96 + 64U \geq 288 \][/tex]
Subtract 96 from both sides to isolate the term with [tex]\(U\)[/tex]:
[tex]\[ 64U \geq 192 \][/tex]
Divide both sides by 64:
[tex]\[ U \geq \frac{192}{64} = 3 \][/tex]
### Conclusion
Therefore, the least number of hours the submarine must travel underwater for Choo Kheng to meet her destination is 3 hours.
Thus, summarizing:
- Surface travel time: 2.6666666666666665 hours,
- Surface travel distance: 96 kilometers,
- Required underwater travel time: 3 hours.
This concludes the solution.
