Question

the function n(x) = 90(.86)^x +69 can be used to predict the temperature of a cup of hot chocolate in degrees Fahrenheit after x minutes. What is the approximate average rate of change of the temperature of the hot chocolate, in degrees per minute, over the interval (0,6)?

Answer 1

The approximate average rate of change of the temperature of the hot chocolate over the interval (0,6) is about -9.165 degrees Fahrenheit per minute.

Step-by-step explanation:

To find the approximate average rate of change of the temperature of the hot chocolate over the interval (0,6), we use the given function n(x) = 90(0.86)^x + 69. We calculate the temperature at both endpoints of the interval and then find the difference between these temperatures divided by the change in time. This will give us the degrees per minute change.

First, we calculate the temperatures at x = 0 and x = 6:

• n(0) = 90(0.86)^0 + 69 = 90 + 69 = 159 degrees Fahrenheit.
• n(6) = 90(0.86)^6 + 69 ≈ 90(0.389) + 69 ≈ 35.01 + 69 ≈ 104.01 degrees Fahrenheit.

Next, we calculate the average rate of change:

Average rate of change = (Temperature at x = 6 - Temperature at x = 0) / (6 - 0)

Average rate of change = (104.01 - 159) / 6

Average rate of change ≈ -54.99 / 6

Average rate of change ≈ -9.165 degrees Fahrenheit per minute