the function ( w(x)=90(0.86)^{x}+69 ) can be used to predict the temperature of a cup of hot chocolate in…

the function ( w(x)=90(0.86)^{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 )?\n1 ( -8.93 )\n2 ( -0.11 )\n3 ( 0.11 )\n4 ( 8.93 )\nsubmit answer

the function ( w(x)=90(0.86)^{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 )?\n1 ( -8.93 )\n2 ( -0.11 )\n3 ( 0.11 )\n4 ( 8.93 )\nsubmit answer

Answer

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}). Here, (a = 0), (b=6), and (f(x)=90(0.86)^{x}+69).

Step2: Calculate (f(0))

Substitute (x = 0) into (f(x)): (f(0)=90(0.86)^{0}+69). Since (a^{0}=1) ((a\neq0)), then (f(0)=90\times1 + 69=159).

Step3: Calculate (f(6))

Substitute (x = 6) into (f(x)): (f(6)=90(0.86)^{6}+69). First, ((0.86)^{6}\approx0.418). Then (f(6)=90\times0.418+69 = 37.62+69=106.62).

Step4: Calculate the average rate of change

Using the formula (\frac{f(6)-f(0)}{6 - 0}=\frac{106.62 - 159}{6}=\frac{-52.38}{6}=- 8.73\approx - 8.93) (there might be some rounding differences in intermediate steps depending on the precision of ((0.86)^{6}) calculation).

Answer:

  1. (-8.93)