Question

find the average rate of change of ( k(x) = 8sqrt{x + 20} ) over the interval (10, 11). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( k(x) \) over the interval \([a, b]\) is given by \(\frac{k(b)-k(a)}{b - a}\). Here, \( a = 10 \), \( b=11 \) and \( k(x)=8\sqrt{x + 20}\).

Step2: Calculate \( k(10) \)

Substitute \( x = 10 \) into \( k(x) \):
\( k(10)=8\sqrt{10 + 20}=8\sqrt{30}\approx8\times5.477 = 43.816\)

Step3: Calculate \( k(11) \)

Substitute \( x = 11 \) into \( k(x) \):
\( k(11)=8\sqrt{11 + 20}=8\sqrt{31}\approx8\times5.568 = 44.544\)

Step4: Calculate the average rate of change

Using the formula \(\frac{k(11)-k(10)}{11 - 10}=\frac{44.544-43.816}{1}= 0.728\approx0.7\) (rounded to the nearest tenth)

Answer:

\(0.7\)