The function \(\mathrm{P(h) = 101.3(0.88)^h}\) models the atmospheric pressure, in kilopascals (kPa), at a certain location, where h is the...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{P(h) = 101.3(0.88)^h}\) models the atmospheric pressure, in kilopascals (kPa), at a certain location, where \(\mathrm{h}\) is the altitude in kilometers (km) above sea level. Which of the following is the best interpretation of the statement \(\mathrm{P(3) \approx 69.1}\) in this context?
- The atmospheric pressure is estimated to be approximately \(\mathrm{3~kPa}\) lower at an altitude of \(\mathrm{3~km}\) than at sea level.
- The atmospheric pressure is estimated to be approximately \(\mathrm{69.1~kPa}\) at an altitude of \(\mathrm{3~km}\) above sea level.
- The altitude is estimated to be approximately \(\mathrm{69.1~km}\) when the atmospheric pressure is \(\mathrm{3~kPa}\).
- The atmospheric pressure is estimated to decrease by approximately \(\mathrm{69.1~kPa}\) for every 3 kilometers of altitude gained.
The atmospheric pressure is estimated to be approximately \(3 \text{ kPa}\) lower at an altitude of \(3 \text{ km}\) than at sea level.
The atmospheric pressure is estimated to be approximately \(69.1 \text{ kPa}\) at an altitude of \(3 \text{ km}\) above sea level.
The altitude is estimated to be approximately \(69.1 \text{ km}\) when the atmospheric pressure is \(3 \text{ kPa}\).
The atmospheric pressure is estimated to decrease by approximately \(69.1 \text{ kPa}\) for every \(3 \text{ kilometers}\) of altitude gained.
1. TRANSLATE the function definition
- Given information:
- \(\mathrm{P(h)}\) = atmospheric pressure in kilopascals (kPa)
- \(\mathrm{h}\) = altitude in kilometers above sea level
- \(\mathrm{P(h) = 101.3(0.88)^h}\) (the specific function)
- What this tells us: \(\mathrm{P(h)}\) is a function where we input altitude and get atmospheric pressure as output
2. TRANSLATE the statement \(\mathrm{P(3) ≈ 69.1}\)
- Breaking down the function notation:
- The input is \(\mathrm{h = 3}\) (this represents 3 km above sea level)
- The output is \(\mathrm{P(3) ≈ 69.1}\) (this represents approximately 69.1 kPa of pressure)
3. INFER the complete interpretation
- Combining input and output meanings:
- "At 3 km above sea level, the pressure is approximately 69.1 kPa"
- This matches choice B exactly
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which number represents the input and which represents the output in function notation.
They might think \(\mathrm{P(3)}\) means "the pressure is 3 kPa" instead of "the pressure when altitude is 3 km." This backwards thinking makes them focus on the wrong numbers and leads to selecting Choice C (The altitude is estimated to be approximately 69.1 km when the atmospheric pressure is 3 kPa) - essentially flipping the input and output.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret the single-point evaluation as representing a rate of change or difference.
Instead of recognizing that \(\mathrm{P(3) ≈ 69.1}\) gives us pressure AT a specific altitude, they think it's telling them something about HOW MUCH pressure changes. This leads them to select Choice D (about pressure decreasing by 69.1 kPa for every 3 km) or Choice A (about pressure being 3 kPa lower).
The Bottom Line:
Function notation problems require careful attention to which number goes where. The key is always: what goes inside the parentheses is your input, and what the function equals is your output.
The atmospheric pressure is estimated to be approximately \(3 \text{ kPa}\) lower at an altitude of \(3 \text{ km}\) than at sea level.
The atmospheric pressure is estimated to be approximately \(69.1 \text{ kPa}\) at an altitude of \(3 \text{ km}\) above sea level.
The altitude is estimated to be approximately \(69.1 \text{ km}\) when the atmospheric pressure is \(3 \text{ kPa}\).
The atmospheric pressure is estimated to decrease by approximately \(69.1 \text{ kPa}\) for every \(3 \text{ kilometers}\) of altitude gained.