A balloon's height h (in meters) changes linearly with time t (in minutes) after observation begins.At t = 0, the...
GMAT Algebra : (Alg) Questions
- A balloon's height \(\mathrm{h}\) (in meters) changes linearly with time \(\mathrm{t}\) (in minutes) after observation begins.
- At \(\mathrm{t = 0}\), the balloon is at a height of 120 meters, and it descends at a constant rate of 3 meters per minute.
- At what time \(\mathrm{t}\) will the balloon reach the ground (that is, when \(\mathrm{h = 0}\))?
Answer Format: Enter your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Balloon starts at height 120 meters at time t = 0
- Height decreases at constant rate of 3 meters per minute
- Height changes linearly with time
- Need to find when height equals 0
- What this tells us: We have a linear function with initial value 120 and slope -3
2. INFER the mathematical approach
- Since height changes linearly, we can write: \(\mathrm{h(t) = 120 - 3t}\)
- 120 is the starting height (y-intercept)
- -3 is the rate of change (slope, negative because descending)
- To find when balloon reaches ground, we need \(\mathrm{h = 0}\)
3. SIMPLIFY by solving the equation
- Set the height function equal to 0:
\(\mathrm{0 = 120 - 3t}\)
- Add 3t to both sides:
\(\mathrm{3t = 120}\)
- Divide both sides by 3:
\(\mathrm{t = 40}\)
Answer: 40
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students might write \(\mathrm{h(t) = 120 + 3t}\) instead of \(\mathrm{h(t) = 120 - 3t}\), thinking "3 meters per minute" means adding rather than recognizing that "descends" means the rate should be negative.
If they use \(\mathrm{h(t) = 120 + 3t}\) and set it equal to 0:
\(\mathrm{0 = 120 + 3t}\)
\(\mathrm{-3t = 120}\)
\(\mathrm{t = -40}\)
This leads to confusion since negative time doesn't make sense in this context, causing them to get stuck and guess.
Second Most Common Error:
Poor INFER reasoning: Students might not realize that "reaches the ground" means \(\mathrm{h = 0}\). They might think they need to find some other relationship or get confused about what the question is asking for.
This leads them to attempt various irrelevant calculations or abandon the systematic approach and guess randomly.
The Bottom Line:
The key challenge is correctly interpreting "descends at 3 meters per minute" as a negative rate of change and understanding that linear word problems require building the function first, then using it to answer the specific question.