A physics student conducts an experiment by throwing a ball straight up from the top of a building. The function...
GMAT Algebra : (Alg) Questions
A physics student conducts an experiment by throwing a ball straight up from the top of a building. The function \(\mathrm{h(t) = -16t^2 + 48t + 120}\) gives the height of the ball, in feet, t seconds after it is thrown. Which of the following is the best interpretation of 120 in this context?
1. TRANSLATE the problem information
- Given: \(\mathrm{h(t) = -16t^2 + 48t + 120}\) represents the height of a ball \(\mathrm{t}\) seconds after being thrown
- Find: What 120 represents in this context
- This is asking us to interpret the constant term in the quadratic function
2. INFER the approach
- To understand what 120 means, we need to think about what happens at \(\mathrm{t = 0}\)
- At \(\mathrm{t = 0}\), we're looking at the initial moment when the ball is thrown
- The value of \(\mathrm{h(0)}\) will tell us the starting height
3. SIMPLIFY by substituting t = 0
- \(\mathrm{h(0) = -16(0)^2 + 48(0) + 120}\)
- \(\mathrm{h(0) = 0 + 0 + 120 = 120}\)
- This means at the initial moment (\(\mathrm{t = 0}\)), the ball is at height 120 feet
4. TRANSLATE back to the context
- Since \(\mathrm{h(0) = 120}\) and \(\mathrm{t = 0}\) is the initial time, 120 represents the initial height of the ball in feet
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may confuse the constant term (120) with the maximum height of the ball's trajectory. They see "120" as a large number and assume it must be the peak height rather than recognizing that the constant term represents the y-intercept or initial value.
This may lead them to select Choice B (The maximum height reached by the ball, in feet)
Second Most Common Error:
Conceptual confusion about quadratic function components: Students may associate 120 with the rate of change because they see other numbers (48, -16) in the function and assume all coefficients relate to rates or slopes, not understanding that the constant term has a different meaning.
This may lead them to select Choice C (The rate of change of the ball's height, in feet per second)
The Bottom Line:
This problem tests whether students understand that in a quadratic function representing a real-world situation, the constant term represents the initial value or starting condition. The key insight is recognizing that evaluating the function at \(\mathrm{t = 0}\) reveals what each parameter means physically.