Question

find the average rate of change of ( k(x) = 10sqrt{x + 11} ) over the interval (-5, 3). write your answer as an integer, fraction, or decimal rounded to the nearest tenth. simplify any fractions.

Explanation:

Step1: Recall the average rate of change formula

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, \( a=-5 \) and \( b = 3 \), and \( k(x)=10\sqrt{x + 11} \).

Step2: Calculate \( k(3) \)

Substitute \( x = 3 \) into \( k(x) \):
\( k(3)=10\sqrt{3 + 11}=10\sqrt{14}\approx10\times3.7417 = 37.417 \)

Step3: Calculate \( k(-5) \)

Substitute \( x=-5 \) into \( k(x) \):
\( k(-5)=10\sqrt{-5 + 11}=10\sqrt{6}\approx10\times2.4495 = 24.495 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{k(3)-k(-5)}{3-(-5)}=\frac{37.417 - 24.495}{3 + 5}=\frac{12.922}{8}\approx1.61525\approx1.6\) (rounded to the nearest tenth)

Answer:

\( 1.6 \)