Question

some water is placed in a bottle inside a refrigerator. suppose the waters temperature (in $^{circ}c$) after $x$ minutes can be modeled with an exponential function. the graph of this function is shown below. use the model to answer the parts to the right. (a) what is the initial temperature of the water? $^{circ}c$ (b) for the first 30 minutes, as time increases, the temperature graph. (c) give the equation of the asymptote. $y=$ choose the statement that best describes the meaning of the asymptote. the water cannot be in the bottle for more than 60 minutes. the temperature of the water does not rise above $15^{circ}c$. the temperature of the water does not fall below $5^{circ}c$.

Explanation:

Step1: Find the initial temperature

The initial temperature is the value of \(y\) when \(x = 0\). Looking at the graph, when \(x=0\) (time \(t = 0\) minutes), the temperature \(y = 60^{\circ}C\).

Step2: Analyze the behavior of the function for part (b)

An exponential function of the form \(y = a\cdot b^{x}+c\) has a horizontal asymptote \(y = c\). As \(x\) (time) increases, the function approaches the horizontal asymptote. From the graph, as \(x\) gets larger (for the first 30 minutes), the temperature approaches \(15^{\circ}C\).

Step3: Determine the equation of the asymptote

The horizontal asymptote of the exponential function (which models the temperature - time relationship) is \(y = 15\).

Step4: Interpret the meaning of the asymptote

The horizontal asymptote \(y = 15\) means that the temperature of the water does not rise above \(15^{\circ}C\). As time goes on (in the context of the first 30 minutes and the long - term behavior of the modeled function), the temperature of the water gets closer and closer to \(15^{\circ}C\) but does not exceed it.

Answer:

(a) \(60\)
(b) approaches \(15^{\circ}C\)
(c) \(y = 15\), The temperature of the water does not rise above \(15^{\circ}C\)