Question

find the average rate of change of ( k(x)=18sqrt{x - 3} ) over the interval ( 5,13 ).
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 \(y = k(x)\) over the interval \([a,b]\) is \(\frac{k(b)-k(a)}{b - a}\). Here, \(a = 5\), \(b=13\), and \(k(x)=18\sqrt{x - 3}\).

Step2: Calculate \(k(13)\) and \(k(5)\)

• For \(x = 13\): \(k(13)=18\sqrt{13 - 3}=18\sqrt{10}\approx18\times3.162 = 56.916\)
• For \(x = 5\): \(k(5)=18\sqrt{5 - 3}=18\sqrt{2}\approx18\times1.414 = 25.452\)

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

\(\frac{k(13)-k(5)}{13 - 5}=\frac{18\sqrt{10}-18\sqrt{2}}{8}=\frac{18(\sqrt{10}-\sqrt{2})}{8}\)

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Answer:

\(3.9\)