The height above ground, H, in meters, of a ball thrown vertically upwards from the top of a building is...
GMAT Advanced Math : (Adv_Math) Questions
The height above ground, \(\mathrm{H}\), in meters, of a ball thrown vertically upwards from the top of a building is modeled by the equation \(\mathrm{H(t) = -5t^2 + 40t + 45}\), where \(\mathrm{t}\) is the time in seconds after the ball is thrown. After how many seconds does the ball reach its maximum height?
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1. INFER the problem type and strategy
- Given information:
- \(\mathrm{H(t) = -5t^2 + 40t + 45}\) (height equation)
- Need to find when ball reaches maximum height
- What this tells us: This is a quadratic function. Since the coefficient of \(\mathrm{t^2}\) is negative (-5), the parabola opens downward, meaning it has a maximum point at its vertex.
2. INFER the appropriate method
- To find when a quadratic function reaches its maximum (or minimum), we find the vertex
- For any quadratic \(\mathrm{at^2 + bt + c}\), the vertex occurs at \(\mathrm{t = -b/(2a)}\)
- This gives us the t-value where the maximum height occurs
3. SIMPLIFY using the vertex formula
- From \(\mathrm{H(t) = -5t^2 + 40t + 45}\), identify the coefficients:
- \(\mathrm{a = -5}\) (coefficient of \(\mathrm{t^2}\))
- \(\mathrm{b = 40}\) (coefficient of t)
- Apply the vertex formula:
\(\mathrm{t = -b/(2a)}\)
\(\mathrm{t = -40/(2(-5))}\)
\(\mathrm{t = -40/(-10)}\)
\(\mathrm{t = 4}\)
Answer: C) 4
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 solve \(\mathrm{H(t) = 0}\) to find when the ball hits the ground, or attempt to take the derivative without understanding that's unnecessary for this level.
This leads to confusion and guessing among the answer choices.
Second Most Common Error:
Poor SIMPLIFY execution: Students apply the vertex formula correctly but make sign errors in the calculation. They might calculate \(\mathrm{t = -40/(2(-5))}\) incorrectly, getting \(\mathrm{t = -4}\) or making other arithmetic mistakes with the negative signs.
Since -4 isn't an answer choice, this leads to confusion and guessing.
The Bottom Line:
This problem tests whether students can identify the mathematical structure (quadratic with maximum at vertex) and apply the correct technique. Many students get stuck because they overthink the physics context instead of focusing on the mathematical relationship.