A ball is launched vertically from a platform. The height h, in meters, of the ball t seconds after launch...

1. TRANSLATE the problem information

• Given function: \(\mathrm{h(t) = -5t^2 + 30t + 2}\)
• This represents height (h) as a function of time (t)
• We need to find when the ball reaches maximum height

2. INFER the mathematical approach

• This is a quadratic function in standard form \(\mathrm{h(t) = at^2 + bt + c}\)
• Since the coefficient of \(\mathrm{t^2}\) is negative (\(\mathrm{a = -5}\)), the parabola opens downward
• A downward-opening parabola has a maximum point at its vertex
• To find maximum height, we need to find the t-value of the vertex

3. TRANSLATE the coefficients

• From \(\mathrm{h(t) = -5t^2 + 30t + 2}\):
  • \(\mathrm{a = -5}\) (coefficient of \(\mathrm{t^2}\))
  • \(\mathrm{b = 30}\) (coefficient of t)
  • \(\mathrm{c = 2}\) (constant term)

4. INFER which formula to use

• For any quadratic \(\mathrm{f(x) = ax^2 + bx + c}\), the vertex occurs at \(\mathrm{x = \frac{-b}{2a}}\)
• This gives us the t-value where maximum height occurs

5. SIMPLIFY using the vertex formula

• \(\mathrm{t = \frac{-b}{2a} = \frac{-30}{2(-5)}}\)
• \(\mathrm{t = \frac{-30}{-10} = 3}\)

Answer: D (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a vertex-finding problem. They might try to set the function equal to zero (\(\mathrm{h(t) = 0}\)) thinking they need to find when the ball hits the ground, or attempt to take the derivative without being in a calculus context.

This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{t = \frac{-30}{2(-5)}}\) but make sign errors in the calculation. They might get \(\mathrm{t = \frac{-30}{-10} = -3}\) (forgetting that negative divided by negative is positive) or \(\mathrm{t = \frac{-30}{10} = -3}\) (missing the negative sign in the denominator).

This may lead them to select an answer that doesn't appear in the choices, causing them to get stuck and guess.

The Bottom Line:

This problem requires recognizing the connection between "maximum height" and "vertex of a parabola." Students who don't make this conceptual leap will struggle to choose the right mathematical tool for the job.