To determine the equation that represents the proportional relationship between the amount of water lost [tex]\( w \)[/tex] and the time [tex]\( h \)[/tex] in hours, let's go through the given situation step-by-step.
1. Understand the given data:
- The pipe is leaking [tex]\(\frac{2}{3}\)[/tex] liters of water every 2 hours.
2. Determine the water loss per hour:
- Since [tex]\(\frac{2}{3}\)[/tex] liters of water are lost in 2 hours, we can find the water loss per hour by dividing [tex]\(\frac{2}{3}\)[/tex] by 2.
[tex]\[
\text{Water loss per hour} = \frac{\frac{2}{3}}{2} = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}
\][/tex]
Thus, the pipe loses [tex]\(\frac{1}{3}\)[/tex] liters of water per hour.
3. Establish the proportional relationship:
- Let [tex]\( w \)[/tex] be the amount of water lost (in liters) and [tex]\( h \)[/tex] be the number of hours.
Since the water loss per hour is [tex]\(\frac{1}{3}\)[/tex] liters, the total amount of water lost [tex]\( w \)[/tex] can be represented as the product of the water loss per hour and the time [tex]\( h \)[/tex] in hours.
[tex]\[
w = \left(\frac{1}{3} \text{ liters per hour}\right) \times h
\][/tex]
[tex]\[
w = \frac{1}{3} h
\][/tex]
4. Match the derived equation with the given options:
- We derived that the relationship is [tex]\( w = \frac{1}{3} h \)[/tex].
Looking at the given options:
- A. [tex]\( w = 3h \)[/tex]
- B. [tex]\( w = \frac{1}{3} h \)[/tex]
- C. [tex]\( h = \frac{1}{3} w \)[/tex]
- D. [tex]\(\frac{2}{3} w = 2h\)[/tex]
The correct equation that matches our derived relationship is:
[tex]\[ \boxed{w = \frac{1}{3} h} \][/tex]
Therefore, the correct answer is option [tex]\( \text{B} \)[/tex].
