A projectile is launched from ground level, and its height h (in feet) above the ground at time t seconds...

1. TRANSLATE the problem information

• Given information:
  • Height function: \(\mathrm{h(t) = (t + 3)(k - t)}\)
  • Maximum height occurs at \(\mathrm{t = 8}\) seconds
  • Need to find: height at \(\mathrm{t = 2}\) seconds
• What this tells us: We have a quadratic function representing projectile motion, and we know when it reaches its peak

2. INFER the mathematical approach

• Key insight: Since this is a quadratic function that opens downward (projectile motion), the maximum occurs at the vertex
• Strategy: Find where the vertex occurs, use the given condition to find k, then calculate h(2)

3. INFER the vertex location using roots

• The function \(\mathrm{h(t) = (t + 3)(k - t)}\) has roots where \(\mathrm{h(t) = 0}\)
• Setting each factor to zero: \(\mathrm{t + 3 = 0}\) gives \(\mathrm{t = -3}\), and \(\mathrm{k - t = 0}\) gives \(\mathrm{t = k}\)
• For any quadratic, the vertex occurs at the midpoint of the roots
• Vertex location: \(\mathrm{t = (-3 + k)/2}\)

4. TRANSLATE the given condition into an equation

• We know the maximum occurs at \(\mathrm{t = 8}\)
• Therefore: \(\mathrm{(-3 + k)/2 = 8}\)

5. SIMPLIFY to find k

• \(\mathrm{(-3 + k)/2 = 8}\)
• Multiply both sides by 2: \(\mathrm{-3 + k = 16}\)
• Add 3 to both sides: \(\mathrm{k = 19}\)

6. SIMPLIFY to find h(2)

• Substitute \(\mathrm{t = 2}\) and \(\mathrm{k = 19}\) into the original function
• \(\mathrm{h(2) = (2 + 3)(19 - 2) = 5 \times 17 = 85}\)

Answer: E) 85

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the maximum of a quadratic occurs at the vertex, or not knowing how to find the vertex location.

Students might try to take the derivative to find the maximum, or they might not understand the relationship between roots and vertex. Without this key insight, they cannot set up the equation to find k, leading to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making arithmetic errors when solving for k or calculating the final answer.

For example, students might correctly set up \(\mathrm{(-3 + k)/2 = 8}\) but then solve incorrectly (perhaps getting \(\mathrm{k = 13}\) instead of \(\mathrm{k = 19}\)), or they might make errors in the final calculation \(\mathrm{h(2) = (2 + 3)(19 - 2)}\). These errors could lead them to select Choice C (75) or another incorrect option.

The Bottom Line:

This problem tests whether students understand the connection between the factored form of a quadratic and its vertex location. The key breakthrough is recognizing that you can find the vertex without expanding the quadratic by using the relationship between roots and vertex position.