A submersible device is used for ocean research. The function \(\mathrm{g(x) = -\frac{1}{55}(x + 19)(x - 35)}\) gives the depth...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = -(1/55)(x + 19)(x - 35)}\) gives depth below surface in meters
  • \(\mathrm{x}\) = minutes after collecting sample, where \(\mathrm{x \gt 0}\)
  • Need to find when submersible reaches surface

• What 'reaching the surface' means mathematically: depth = 0, so \(\mathrm{g(x) = 0}\)

2. INFER the approach

• To find when \(\mathrm{g(x) = 0}\), substitute 0 for \(\mathrm{g(x)}\) in the function
• This creates an equation we can solve for \(\mathrm{x}\)

3. Set up and SIMPLIFY the equation

• Start with: \(\mathrm{0 = -(1/55)(x + 19)(x - 35)}\)
• Multiply both sides by -55: \(\mathrm{0 = (x + 19)(x - 35)}\)

4. INFER solution method using zero product property

• Since we have a product equal to zero, either factor must equal zero
• Set each factor to zero:
  • \(\mathrm{x + 19 = 0}\), which gives \(\mathrm{x = -19}\)
  • \(\mathrm{x - 35 = 0}\), which gives \(\mathrm{x = 35}\)

5. APPLY CONSTRAINTS to select final answer

• Since \(\mathrm{x \gt 0}\) (given constraint), we reject \(\mathrm{x = -19}\)
• Therefore \(\mathrm{x = 35}\)

Answer: 35

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not connect 'reaches the surface' with 'depth = 0'

Some students might think the surface corresponds to the maximum depth or get confused about what the function represents. They may try to find when \(\mathrm{g(x)}\) is maximized rather than when \(\mathrm{g(x) = 0}\). This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly find \(\mathrm{x = -19}\) and \(\mathrm{x = 35}\) but forget to apply the constraint \(\mathrm{x \gt 0}\)

They might select both solutions or randomly pick \(\mathrm{x = -19}\), not realizing that negative time doesn't make sense in this context. This may lead them to select -19 if it were an answer choice, or causes confusion about which solution is correct.

The Bottom Line:

This problem requires students to connect real-world language ('reaches surface') with mathematical conditions (function equals zero), then systematically solve while respecting contextual constraints. The key insight is recognizing that 'surface' means zero depth.