Question

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 )?
1 ( -8.93 )
2 ( -0.11 )
3 ( 0.11 )
4 ( 8.93 )
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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 \(a^{0}=1\) (\(a
eq0\)), 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. \(-8.93\)