\(\mathrm{h(t) = -16t^2 + b}\)The function h estimates an object's height, in feet, above the ground t seconds after the...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{h(t) = -16t^2 + b}\)
The function \(\mathrm{h}\) estimates an object's height, in feet, above the ground \(\mathrm{t}\) seconds after the object is dropped, where \(\mathrm{b}\) is a constant. The function estimates that the object is 3,364 feet above the ground when it is dropped at \(\mathrm{t = 0}\). Approximately how many seconds after being dropped does the function estimate the object will hit the ground?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{h(t) = -16t^2 + b}\) (height function)
- At \(\mathrm{t = 0}\), object is 3,364 feet above ground
- Need to find when object hits ground (height = 0)
2. TRANSLATE the initial condition to find b
- When \(\mathrm{t = 0}\): \(\mathrm{h(0) = 3,364}\)
- Substituting: \(\mathrm{h(0) = -16(0)^2 + b = b}\)
- Therefore: \(\mathrm{b = 3,364}\)
3. INFER the complete function and next step
- Complete function: \(\mathrm{h(t) = -16t^2 + 3,364}\)
- Key insight: "hits the ground" means height = 0
- Strategy: Set \(\mathrm{h(t) = 0}\) and solve for t
4. SIMPLIFY the equation step by step
- Set up equation: \(\mathrm{0 = -16t^2 + 3,364}\)
- Rearrange: \(\mathrm{16t^2 = 3,364}\)
- Divide by 16: \(\mathrm{t^2 = 210.25}\)
- Take square root: \(\mathrm{t = \sqrt{210.25} \approx 14.50}\) (use calculator)
5. APPLY CONSTRAINTS for final answer
- Since time after being dropped must be positive: \(\mathrm{t = 14.50}\) seconds
Answer: B. 14.50
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students may try to work directly with \(\mathrm{h(t) = -16t^2 + b}\) without first finding the value of b from the initial condition. They might set \(\mathrm{-16t^2 + b = 0}\) and get stuck because they don't know what b equals.
This leads to confusion and abandoning systematic solution, resulting in guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up the equation \(\mathrm{16t^2 = 3,364}\) but make calculation errors. They might incorrectly compute \(\mathrm{3,364 \div 16}\) or make errors when approximating \(\mathrm{\sqrt{210.25}}\).
Common calculation mistakes could lead them to select Choice A (7.25) if they take the square root of half the correct value, or other incorrect choices.
The Bottom Line:
This problem requires careful attention to initial conditions and systematic equation solving. The key insight is recognizing that you must use the given information at \(\mathrm{t = 0}\) to find the unknown constant before you can solve for when the object hits the ground.