A tank initially contains 260 liters of water. Water flows into the tank at a constant rate, so the amount...

1. TRANSLATE the problem information

• Given information:
  • Initial water: 260 liters at t = 0
  • Water flows at constant rate (linear function)
  • After 68 minutes: 600 liters total
  • Find: Amount after 350 minutes

• What this tells us: Since the rate is constant, W(t) follows a linear pattern

2. INFER the mathematical approach

• Since water flows at a constant rate, this is a linear function problem
• Linear functions have the form: \(\mathrm{W(t) = rt + b}\)
• Here, \(\mathrm{r}\) = inflow rate (liters/minute) and \(\mathrm{b}\) = initial amount
• So our function is: \(\mathrm{W(t) = rt + 260}\)

3. SIMPLIFY to find the rate

• Use the known data point \(\mathrm{W(68) = 600}\):
  \(\mathrm{600 = 68r + 260}\)
  \(\mathrm{68r = 340}\)
  \(\mathrm{r = 5}\) liters per minute

• Our complete function: \(\mathrm{W(t) = 5t + 260}\)

4. SIMPLIFY to find the final answer

• Calculate \(\mathrm{W(350)}\):
  \(\mathrm{W(350) = 5(350) + 260}\)
  \(\mathrm{= 1750 + 260}\)
  \(\mathrm{= 2010}\)

Answer: 2010

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may misinterpret what "260 liters initially" means in the context of a linear function. They might think the function starts from zero and add 260 later, setting up \(\mathrm{W(t) = rt}\) instead of \(\mathrm{W(t) = rt + 260}\).

With \(\mathrm{W(t) = rt}\), using \(\mathrm{W(68) = 600}\) gives \(\mathrm{r = 600/68 \approx 8.82}\). Then \(\mathrm{W(350) = 8.82 \times 350 \approx 3087}\), leading to an incorrect answer.

Second Most Common Error:

Poor INFER strategy: Students might not recognize this as a linear function problem at all. They may try to find some complex relationship or get confused about what "constant rate" means, leading to overcomplicated approaches that don't utilize the linear function structure.

This leads to confusion and abandoning systematic solution, resulting in guessing.

The Bottom Line:

This problem tests whether students can translate a real-world scenario into proper linear function notation, particularly understanding that the y-intercept represents the initial condition, not something to be added later.