Question

find the average rate of change of ( k(x)=5sqrt{x}+8 ) over the interval ( 1,16 ). 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 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, \(a = 1\), \(b=16\), and \(f(x)=5\sqrt{x}+8\).

Step2: Calculate \(f(1)\) and \(f(16)\)

• For \(x = 1\): \(f(1)=5\sqrt{1}+8=5\times1 + 8=13\).
• For \(x = 16\): \(f(16)=5\sqrt{16}+8=5\times4+8=20 + 8=28\).

Step3: Substitute into the average - rate - of - change formula

\(\frac{f(16)-f(1)}{16 - 1}=\frac{28 - 13}{15}=\frac{15}{15}=1\).

Answer:

\(1\)