a) use the exponential function to estimate the temperature difference y when 30 minutes have elapsed. Report estimated temperature difference to the nearest tenth of a degreeb) since y=C-69, we have coffee temperature C=y+69. Take the difference estimate from part (a) and add 69 degrees. Fill in the blank: When 25 minutes have elapsed, the estimated coffee temperature is ____ degrees.c) suppose the coffee temperature C is 100 degrees. Then y=C-69= ____ degrees is the temperature difference between the coffee and room temperatures d) consider the equation ___=90e-0.023t where the ___ is filled in with the answer from part ce) Solve part d equation for t, to the nearest tenth **Data Summary:**A cup of hot coffee was placed in a room maintained at a constant temperature of 69 degrees Fahrenheit. The coffee temperature was recorded periodically, as shown in Table 1 below.**Table 1: Coffee Temperature Over Time**| Time Elapsed (minutes) (t) | Coffee Temperature (degrees F) (C) ||----------------------------|------------------------------------|| 0 | 166.0 || 10 | 140.5 || 20 | 125.2 || 30 | 110.3 || 40 | 104.5 || 50 | 98.4 || 60 | 93.9 |**Remarks:**Common sense suggests that the coffee will cool off, and its temperature will decrease, approaching the ambient room temperature of 69 degrees.Thus, the temperature difference between the coffee temperature and the room temperature will decrease to zero.We will fit the temperature difference data (Table 2) to an exponential curve of the form $ y = A e^{-bt} $.As $ t $ gets larger, $ y $ will get closer and closer to zero, reflecting the behavior of the temperature difference.To analyze the data where $ t $ represents the time elapsed and $ y = C - 69 $—the temperature difference between the coffee temperature and the room temperature—refer to the following table:**Table 2: Temperature Difference Over Time**| Time Elapsed (minutes) (t) | Temperature Difference (degrees F) (y = C - 69) ||----------------------------|-------------------------------------------------| | 0 | 97.0 || 10 | 71.5 || 20 | 56.2 || 30 | 41.3 || 40 | 35.5 || 50 | 29.4 || 60 | 24.9 | The graph illustrates the "Temperature Difference between Coffee and Room" over time. The x-axis represents the Time Elapsed in minutes, ranging from 0 to 70 minutes. The y-axis shows the Temperature Difference in degrees, spanning from 0 to 120 degrees.Data points on the graph are plotted and connected by a smooth curve, indicating the decrease in temperature difference as time progresses. The relationship is modeled by the equation:\[ y = 90 e^{-0.023t} \]where $ t $ denotes the time elapsed in minutes, and $ y $ is the temperature difference in degrees. The graph also shows an $ R^2 $ value of 0.985, suggesting a strong fit of the data to the exponential model. As time increases, the temperature difference decreases exponentially, indicating that the coffee cools down over time until it approaches room temperature.

Name: a) use the exponential function to estimate the temperature difference
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: a) use the exponential function to estimate the temperature difference y when 30 minutes have elapsed. Report estimated temperature difference to the nearest tenth of a degreeb) since y=C-69, we have coffee temperature C=y+69. Take the difference est