\(\mathrm{f(t) = 55t - 2t^2}\) The function f is defined by the given equation. The function g is defined by...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(t) = 55t - 2t^2}\)
  • \(\mathrm{g(t) = f(t) + 3}\)
  • Need to find maximum value of \(\mathrm{g(t)}\)

2. INFER the approach needed

• Since \(\mathrm{g(t)}\) will be quadratic with negative leading coefficient, it has a maximum at its vertex
• To find the maximum value, I need to rewrite \(\mathrm{g(t)}\) in vertex form using completing the square
• First, I'll substitute to get the full expression for \(\mathrm{g(t)}\)

3. SIMPLIFY to find g(t)

• \(\mathrm{g(t) = f(t) + 3 = (55t - 2t^2) + 3 = -2t^2 + 55t + 3}\)

4. SIMPLIFY by completing the square

• Start: \(\mathrm{g(t) = -2t^2 + 55t + 3}\)
• Factor out -2 from first two terms: \(\mathrm{g(t) = -2(t^2 - (55/2)t) + 3}\)
• Complete the square inside parentheses:
  • Take half of coefficient: \(\mathrm{(55/2) ÷ 2 = 55/4}\)
  • Square it: \(\mathrm{(55/4)^2}\)
  • So: \(\mathrm{t^2 - (55/2)t = (t - 55/4)^2 - (55/4)^2}\)

5. SIMPLIFY to get vertex form

• Substitute back: \(\mathrm{g(t) = -2[(t - 55/4)^2 - (55/4)^2] + 3}\)
• Distribute: \(\mathrm{g(t) = -2(t - 55/4)^2 + 2(55/4)^2 + 3}\)
• This is vertex form: \(\mathrm{g(t) = -2(t - 55/4)^2 + [2(55/4)^2 + 3]}\)

6. INFER the maximum value

• Since coefficient of squared term is negative (-2), the vertex gives the maximum
• Maximum occurs when \(\mathrm{(t - 55/4)^2 = 0}\)
• Maximum value = \(\mathrm{2(55/4)^2 + 3 = 3 + 2(55/4)^2}\)

Answer: B. \(\mathrm{3 + 2(55/4)^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors or arithmetic mistakes during completing the square process

Students often struggle with the multiple algebraic steps, particularly when factoring out the negative coefficient and distributing back. They might get: \(\mathrm{-2(t - 55/4)^2 - 2(55/4)^2 + 3}\) instead of the correct \(\mathrm{+2(55/4)^2}\). This leads them to select Choice C (\(\mathrm{3 - 2(55/4)^2}\)) or causes confusion about which operations to perform.

Second Most Common Error:

Inadequate INFER reasoning: Not recognizing that completing the square is needed to find the maximum

Some students try to use the vertex formula \(\mathrm{t = -b/(2a)}\) to find the t-coordinate, then substitute back to find the maximum value. While this approach can work, it's more computation-heavy and prone to arithmetic errors. Students may calculate incorrectly and end up with expressions that don't match any answer choice, leading to guessing.

The Bottom Line:

This problem requires systematic algebraic manipulation through completing the square. The key insight is recognizing that all answer choices are in the form "3 + something" or "3 - something," which should guide students toward the vertex form approach where the constant term clearly appears.