Data Analysis An object at a temperature of 160^∘ C was removed from a furnace and placed in a r

step 1 Notice that when you're observing the graph, there's a horizontal asymptote and it corresponds t

Data Analysis An object at a temperature of 160^∘ C was...

Data Analysis An object at a temperature of $160^{\circ} \mathrm{C}$ was removed from a furnace and placed in a room at $20^{\circ} \mathrm{C}$. The temperature $T$ of the object was measured each hour $h$ and recorded in the table. A model for the data is given by $T=20\left[1+7\left(2^{-h}\right)\right] .$ The figure shows the graph of this model. $$ \begin{array}{|c|c|} \hline \text { Tramples } & \text { Temples } \\ \hline 0 & 160^{\circ} \\ 1 & 90^{\circ} \\ 2 & 56^{\circ} \\ 3 & 38^{\circ} \\ 4 & 29^{\circ} \\ 5 & 24^{\circ} \\ \hline \end{array} $$ (a) Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. (b) Use the model to approximate the time when the temperature of the object was $100^{\circ} \mathrm{C}$

Data Analysis An object at a temperature of $160^{\circ} \mathrm{C}$ was removed from a furnace and placed in a room at $20^{\circ} \mathrm{C}$. The temperature $T$ of the object was measured each hour $h$ and recorded in the table. A model for the data is given by $T=20\left[1+7\left(2^{-h}\right)\right] .$ The figure shows the graph of this model.
$$
\begin{array}{|c|c|} \hline \text { Tramples } & \text { Temples } \\ \hline 0 & 160^{\circ} \\ 1 & 90^{\circ} \\ 2 & 56^{\circ} \\ 3 & 38^{\circ} \\ 4 & 29^{\circ} \\ 5 & 24^{\circ} \\ \hline \end{array}
$$
(a) Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem.
(b) Use the model to approximate the time when the temperature of the object was $100^{\circ} \mathrm{C}$

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Key Concepts

Exponential Decay Model

Exponential decay models describe processes in which a quantity decreases at a rate proportional to its current value. They are widely used in various fields, including physics and engineering, to model phenomena such as cooling of objects, radioactive decay, and depreciation. The model typically involves an exponential function that indicates how quickly the quantity approaches a lower limit over time.

Horizontal Asymptote

A horizontal asymptote is a line that the graph of a function approaches as the independent variable (often time) goes to infinity. In the context of a cooling object, the horizontal asymptote represents the ambient temperature of the surroundings, which the object approaches as it cools over time, indicating that despite ongoing changes, the temperature will never fall below this ambient value.

Logarithmic Equation Solving

Solving exponential equations often involves the use of logarithms, which allow you to isolate the variable in the exponent. This is essential when determining the time at which a given condition is met in exponential models. By applying logarithms to both sides of an equation, you can solve for the unknown exponent, providing a systematic approach to resolving variables in growth or decay models.

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