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Answer :
Certainly! Let's analyze the function that models the cooling of the warm object:
The function given is [tex]\(\pi(t) = 249 e^{-0.035 t} + 82\)[/tex].
This function describes how the temperature of the object changes over time [tex]\(t\)[/tex]. It is composed of two parts:
1. The term [tex]\(249 e^{-0.035 t}\)[/tex]: This is an exponential decay function. As time [tex]\(t\)[/tex] increases and approaches infinity, the exponential term [tex]\(e^{-0.035 t}\)[/tex] decreases and approaches zero. This means that the influence of this term on the temperature becomes negligible over time.
2. The constant term [tex]\(82\)[/tex]: This represents the environmental temperature or the temperature at which the object will eventually stabilize once the effect of the exponential term has diminished.
Now, let's determine the temperature at which the object will stabilize:
- As [tex]\(t\)[/tex] approaches infinity, the term [tex]\(249 e^{-0.035 t}\)[/tex] approaches zero.
- Therefore, the function simplifies to [tex]\(\pi(t) = 82\)[/tex].
Thus, the constant temperature, in [tex]\({ }^{\circ} F\)[/tex], at which the object's temperature will eventually stabilize is [tex]\(82^{\circ} F\)[/tex].
The function given is [tex]\(\pi(t) = 249 e^{-0.035 t} + 82\)[/tex].
This function describes how the temperature of the object changes over time [tex]\(t\)[/tex]. It is composed of two parts:
1. The term [tex]\(249 e^{-0.035 t}\)[/tex]: This is an exponential decay function. As time [tex]\(t\)[/tex] increases and approaches infinity, the exponential term [tex]\(e^{-0.035 t}\)[/tex] decreases and approaches zero. This means that the influence of this term on the temperature becomes negligible over time.
2. The constant term [tex]\(82\)[/tex]: This represents the environmental temperature or the temperature at which the object will eventually stabilize once the effect of the exponential term has diminished.
Now, let's determine the temperature at which the object will stabilize:
- As [tex]\(t\)[/tex] approaches infinity, the term [tex]\(249 e^{-0.035 t}\)[/tex] approaches zero.
- Therefore, the function simplifies to [tex]\(\pi(t) = 82\)[/tex].
Thus, the constant temperature, in [tex]\({ }^{\circ} F\)[/tex], at which the object's temperature will eventually stabilize is [tex]\(82^{\circ} F\)[/tex].
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