Question

benchmark 1 spiral
the volume of a cube is ( v(s)=s^{3} ), where ( s ) is the side length (in ( mathrm{cm} )).
find the average rate of change of volume as ( s ) increases from ( 2 mathrm{~cm} ) to ( 5 mathrm{~cm} ).
39
50
3
117

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x=a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(f(s)=s^{3}\), \(a = 2\), and \(b = 5\).

Step2: Calculate \(f(5)\) and \(f(2)\)

• For \(s = 5\), \(f(5)=5^{3}=125\).
• For \(s = 2\), \(f(2)=2^{3}=8\).

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

\(\frac{f(5)-f(2)}{5 - 2}=\frac{125 - 8}{3}\).
First, calculate the numerator: \(125-8 = 117\).
Then, divide by the denominator: \(\frac{117}{3}=39\).

Answer:

39