A glass filled with ice and water is set on a table in a room at a temperature of 71 Fahrenheit. After placing a emperat

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A Glass Filled With Ice And Water Is Set On A Table In A Room At A Temperature Of 71 Fahrenheit After Placing A Emperat 1 (169.95 KiB) Viewed 36 times

A glass filled with ice and water is set on a table in a room at a temperature of 71 Fahrenheit. After placing a emperature probe in the glass, we record the following temperatures at time t (in minutes). a) Consider the function F(t)=a−be−kt, where a,b, and k are positive constants. Carcfully describe F as a transformation of the parent function f(t)=e′. In addition, why is F a reasonable model to choose to represent the temperature of the water in the glass at time t ? b) In terms of the constants in the problem (a,b, and /or k), what will be the long-range behavior of F(t)=a−be−kt ? In terms of the temperature of the water in the glass and the surrounding room, what should we expect will be the long-range behavior of the temperature of the water? What constant(s) among a,b, and k can we now determine'? Explain your thinking and find the constant's value. c) In terms of the constants a,b, and or k in our model F(t)=a−be−kt, what is the value of F(0) ? What is the temperature of the water in the glass at time t=0 ? What constant(s) among a,b, and k can we now determinc'? Explain your thinking and find the second constant's value. d) At this point, we have determined two of the three constants and have used two of the three data points (the long-range behavior is acting like a data point) connected to temperature. Use the remaining data point, the two constants we know, and the form of the model (F(t)=a−be−kt) to find the remaining constant's valuc. Find it exactly using algebraic work that you show or describe in detail, and then state an approximation that is accurate to three decimal places. Make sure to state the model with all three constants a,b, and k included. c) Find the EXACT time the water in the glass reaches 60 , according to the model. Clearly show your algebraic work and thinking, and then report the value accurate to three decimal places. f) Find the average rate of change of F on the intervals [5,8] and [8,11]. Explain how your values tell you the shape (increasing/decreasing and concave up/concave down) of the function. g) In an appropriate window, plot the model F(t), the given data points, the point (,60) that you found in part (c), and the horizontal asymptotc. Include this plot in your write-up.