The function \(\mathrm{T(h) = 95(0.92)^h}\) models the temperature, in degrees Fahrenheit, of a cup of coffee h hours after it...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{T(h) = 95(0.92)^h}\) models the temperature, in degrees Fahrenheit, of a cup of coffee h hours after it was brewed. Which of the following is the best interpretation of '\(\mathrm{T(3) \approx 74}\)' in this context?
- The temperature of the coffee is estimated to be approximately 3 degrees lower after 3 hours than when it was brewed.
- The temperature of the coffee is estimated to be approximately 74 degrees Fahrenheit after 3 hours.
- The temperature of the coffee is estimated to be approximately three-fourths of its original temperature after 3 hours.
- The temperature of the coffee is estimated to decrease by approximately 74 degrees every 3 hours.
The temperature of the coffee is estimated to be approximately 3 degrees lower after 3 hours than when it was brewed.
The temperature of the coffee is estimated to be approximately 74 degrees Fahrenheit after 3 hours.
The temperature of the coffee is estimated to be approximately three-fourths of its original temperature after 3 hours.
The temperature of the coffee is estimated to decrease by approximately 74 degrees every 3 hours.
1. TRANSLATE the function components
- Given: \(\mathrm{T(h) = 95(0.92)^h}\) models coffee temperature
- Key translations:
- \(\mathrm{h}\) = hours after the coffee was brewed
- \(\mathrm{T(h)}\) = temperature (in °F) at time \(\mathrm{h}\)
- \(\mathrm{T(3)}\) = temperature after 3 hours specifically
2. TRANSLATE the given statement
- "T(3) is approximately equal to 74"
- This means: \(\mathrm{T(3) ≈ 74}\)
- In context: The temperature after 3 hours is approximately 74 degrees Fahrenheit
3. INFER what this tells us about the coffee
- Since \(\mathrm{T(3)}\) represents the output value when \(\mathrm{h = 3}\)
- And the output represents temperature in degrees Fahrenheit
- Therefore: After 3 hours, the coffee has cooled to about 74°F
4. Verify our interpretation makes sense
- Initial temperature: \(\mathrm{T(0) = 95(0.92)^0 = 95°F}\)
- After 3 hours: \(\mathrm{T(3) = 95(0.92)^3 ≈ 74°F}\)
- This shows realistic cooling from 95°F to 74°F over 3 hours
Answer: B. The temperature of the coffee is estimated to be approximately 74 degrees Fahrenheit after 3 hours.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what \(\mathrm{T(3) = 74}\) represents, focusing on the number "3" in the input rather than understanding it as the time value.
They might think "3 degrees" appears somewhere in the answer because they see "T(3)" and assume the 3 refers to a temperature change. This confusion about function notation leads them to select Choice A (3 degrees lower).
Second Most Common Error:
Poor INFER reasoning: Students recognize that 74 is less than 95, so they calculate \(\mathrm{95 - 74 = 21}\), but then incorrectly think this 21-degree drop happens repeatedly.
Without understanding exponential decay, they assume linear decrease and think the temperature drops by some amount every 3 hours. This may lead them to select Choice D (decreases by 74 degrees every 3 hours).
The Bottom Line:
This problem tests whether students truly understand function notation in context. The key insight is that \(\mathrm{T(3) = 74}\) gives you the actual temperature value at a specific time, not information about temperature changes or rates of change.
The temperature of the coffee is estimated to be approximately 3 degrees lower after 3 hours than when it was brewed.
The temperature of the coffee is estimated to be approximately 74 degrees Fahrenheit after 3 hours.
The temperature of the coffee is estimated to be approximately three-fourths of its original temperature after 3 hours.
The temperature of the coffee is estimated to decrease by approximately 74 degrees every 3 hours.