Question

let ( f(t) ) be the temperature of a cup of coffee ( t ) minutes after it has been poured. interpret ( f(9)=130 ) and ( f^{prime}(9)=-7 ). estimate the temperature of the coffee after 9 minutes and 36 seconds, that is, after 9.6 minutes.
what does ( f(9)=130 ) imply?
a. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is 9 degrees.
b. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 130 degrees per minute.
c. 130 minutes after the coffee has been poured, the temperature of the cup of coffee is rising at a rate of 9 degrees per minute.
d. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.

Explanation:

Step1: Analyze the function \( f(t) \)

The function \( f(t) \) represents the temperature of the coffee \( t \) minutes after it has been poured. When \( t = 9 \), \( f(9) \) gives the temperature at \( t = 9 \) minutes.

Step2: Analyze each option

• Option A: Incorrect. The input \( t \) is minutes, and \( f(t) \) is temperature. Here \( t = 9 \) (not 130) and \( f(t)=130 \) (not 9).
• Option B: Incorrect. \( f(9) = 130 \) is the value of the function (temperature), not the derivative (rate of change). The derivative \( f^{\prime}(t) \) represents the rate of change of temperature.
• Option C: Incorrect. Similar to Option A, the values of \( t \) and \( f(t) \) are mis - matched.
• Option D: Correct. Since \( t = 9 \) (minutes) and \( f(t) \) is the temperature function, \( f(9)=130 \) means 9 minutes after pouring, the temperature is 130 degrees.

Answer:

D. 9 minutes after the coffee has been poured, the temperature of the cup of coffee is 130 degrees.