Sarah brewed a cup of coffee and immediately measured its temperature. The function \(\mathrm{h(m) = 180(0.88)^m}\), where 0 leq m...
GMAT Advanced Math : (Adv_Math) Questions
Sarah brewed a cup of coffee and immediately measured its temperature. The function \(\mathrm{h(m) = 180(0.88)^m}\), where \(\mathrm{0 \leq m \leq 30}\), gives the predicted temperature, in degrees Fahrenheit, of the coffee m minutes after it was brewed. What is the best interpretation of the statement '\(\mathrm{h(15)}\) is approximately equal to \(\mathrm{21}\)' in this context?
When the coffee's predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit, it is \(\mathrm{15\%}\) lower than the predicted temperature \(\mathrm{1}\) minute earlier.
When the coffee's predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit, it is \(\mathrm{15}\) times lower than the initial temperature when the coffee was brewed.
From when Sarah brewed the coffee to \(\mathrm{15}\) minutes after she brewed it, the coffee's predicted temperature decreased by a total of approximately \(\mathrm{21}\) degrees Fahrenheit.
\(\mathrm{15}\) minutes after Sarah brewed the coffee, its predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit.
1. TRANSLATE the mathematical statement
- Given: \(\mathrm{h(15)}\) is approximately equal to 21
- The function is \(\mathrm{h(m) = 180(0.88)^m}\) where \(\mathrm{m}\) represents minutes after brewing
- TRANSLATE this to: "When we plug in \(\mathrm{m = 15}\), we get approximately 21"
2. INFER what this means in context
- Since \(\mathrm{h(m)}\) gives the temperature \(\mathrm{m}\) minutes after brewing
- \(\mathrm{h(15)}\) must give the temperature 15 minutes after brewing
- If \(\mathrm{h(15) \approx 21}\), then the temperature 15 minutes after brewing is approximately 21°F
3. TRANSLATE this understanding to the answer choices
- Look for the choice that says: "15 minutes after brewing, temperature is 21°F"
- Choice (D) states exactly this interpretation
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what the numbers in "\(\mathrm{h(15) \approx 21}\)" represent in context.
Some students think the "15" refers to a temperature change or percentage, rather than the input variable (time). They might focus on the 15% mentioned in choice (A) and think this connects to the 15 in \(\mathrm{h(15)}\). Others confuse which number represents time versus temperature, thinking 21 might represent minutes instead of degrees.
This may lead them to select Choice (A) (15% temperature decrease) or get confused and guess.
Second Most Common Error:
Weak INFER skill: Students don't connect function evaluation to its contextual meaning.
They might recognize \(\mathrm{h(15) = 21}\) mathematically but fail to interpret what this means in the coffee cooling scenario. They see the equals sign as just a mathematical relationship rather than understanding it describes the temperature at a specific moment in time.
This causes them to get stuck and randomly select an answer.
The Bottom Line:
This problem tests whether students can bridge between abstract function notation and real-world interpretation. The key insight is recognizing that \(\mathrm{h(15)}\) represents "plugging 15 into the function" which gives the temperature at that specific time point.
When the coffee's predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit, it is \(\mathrm{15\%}\) lower than the predicted temperature \(\mathrm{1}\) minute earlier.
When the coffee's predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit, it is \(\mathrm{15}\) times lower than the initial temperature when the coffee was brewed.
From when Sarah brewed the coffee to \(\mathrm{15}\) minutes after she brewed it, the coffee's predicted temperature decreased by a total of approximately \(\mathrm{21}\) degrees Fahrenheit.
\(\mathrm{15}\) minutes after Sarah brewed the coffee, its predicted temperature is approximately \(\mathrm{21}\) degrees Fahrenheit.