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Answer :
Certainly! Let's explore how the temperature of the object behaves over time according to the given cooling function:
The function is [tex]\( \pi(t) = 249 e^{-0.035 t} + 82 \)[/tex].
1. Understand the Components of the Function:
- The function describes how the temperature, [tex]\( \pi(t) \)[/tex], changes over time [tex]\( t \)[/tex].
- [tex]\( 249 e^{-0.035 t} \)[/tex] is a term that depends on time. It represents how much the temperature is above a certain baseline temperature.
- [tex]\( 82 \)[/tex] is a constant. It represents the environmental temperature where the object is cooling towards.
2. Analyze the Behavior as Time Increases:
- As time [tex]\( t \)[/tex] becomes very large (approaches infinity), the expression [tex]\( e^{-0.035 t} \)[/tex] becomes very small.
- Specifically, [tex]\( e^{-0.035 t} \)[/tex] approaches 0 because exponents with negative powers become tiny as the exponent grows.
3. Evaluate the Function at Infinity:
- As a result of the above, the term [tex]\( 249 e^{-0.035 t} \)[/tex] approaches 0 as [tex]\( t \)[/tex] gets larger.
- Therefore, the effect of this term on [tex]\( \pi(t) \)[/tex] diminishes over time.
4. Identify the Stabilizing Temperature:
- What's left in the function as the first term vanishes is the constant [tex]\( 82 \)[/tex].
- Hence, the temperature that the object is cooling towards, which becomes the stabilized temperature, is the constant term in the function.
In conclusion, the constant temperature where the object's temperature will eventually stabilize is [tex]\( \boxed{82^{\circ} F} \)[/tex].
The function is [tex]\( \pi(t) = 249 e^{-0.035 t} + 82 \)[/tex].
1. Understand the Components of the Function:
- The function describes how the temperature, [tex]\( \pi(t) \)[/tex], changes over time [tex]\( t \)[/tex].
- [tex]\( 249 e^{-0.035 t} \)[/tex] is a term that depends on time. It represents how much the temperature is above a certain baseline temperature.
- [tex]\( 82 \)[/tex] is a constant. It represents the environmental temperature where the object is cooling towards.
2. Analyze the Behavior as Time Increases:
- As time [tex]\( t \)[/tex] becomes very large (approaches infinity), the expression [tex]\( e^{-0.035 t} \)[/tex] becomes very small.
- Specifically, [tex]\( e^{-0.035 t} \)[/tex] approaches 0 because exponents with negative powers become tiny as the exponent grows.
3. Evaluate the Function at Infinity:
- As a result of the above, the term [tex]\( 249 e^{-0.035 t} \)[/tex] approaches 0 as [tex]\( t \)[/tex] gets larger.
- Therefore, the effect of this term on [tex]\( \pi(t) \)[/tex] diminishes over time.
4. Identify the Stabilizing Temperature:
- What's left in the function as the first term vanishes is the constant [tex]\( 82 \)[/tex].
- Hence, the temperature that the object is cooling towards, which becomes the stabilized temperature, is the constant term in the function.
In conclusion, the constant temperature where the object's temperature will eventually stabilize is [tex]\( \boxed{82^{\circ} F} \)[/tex].
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