The table below shows several values of t and the corresponding values of h determined by the equation h =...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{h = -5t^2 + 40t + 10}\)
  • Table shows that when \(\mathrm{t = 3}\), \(\mathrm{h = k}\)
  • Need to find the value of k

• What this tells us: We need to substitute \(\mathrm{t = 3}\) into the equation to find k

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{t = 3}\) into \(\mathrm{h = -5t^2 + 40t + 10}\):
  \(\mathrm{k = -5(3)^2 + 40(3) + 10}\)

• Follow order of operations carefully:
  • First, calculate the exponent: \(\mathrm{3^2 = 9}\)
  • \(\mathrm{k = -5(9) + 40(3) + 10}\)
  • Next, perform multiplications: \(\mathrm{-5(9) = -45}\) and \(\mathrm{40(3) = 120}\)
  • \(\mathrm{k = -45 + 120 + 10}\)
  • Finally, add from left to right: \(\mathrm{-45 + 120 = 75}\), then \(\mathrm{75 + 10 = 85}\)

Answer: B (85)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes during the final addition step

After correctly computing \(\mathrm{-5(9) = -45}\) and \(\mathrm{40(3) = 120}\), they might incorrectly calculate \(\mathrm{-45 + 120 = 85}\) (instead of 75), leading to \(\mathrm{k = 85 + 10 = 95}\).

This may lead them to select Choice C (95)

Second Most Common Error:

Incomplete SIMPLIFY process: Students forget to include the constant term

They correctly compute \(\mathrm{-5(9) + 40(3) = -45 + 120 = 75}\), but forget to add the final +10 term from the original equation.

This may lead them to select Choice A (75)

The Bottom Line:

This problem tests careful execution of order of operations and attention to all terms in the equation. Success requires systematic work through each step without rushing the arithmetic.