the function $f(x)=2.5x^{2}+21x + 36$ models the dimensions of a rectangular piece of art. what is the…

Answer:

First, recall the formula for the average rate of change of a function (y = f(x)) over the interval ([a,b]) which is (\frac{f(b)-f(a)}{b - a}). Here, (a = 5), (b=10), and (f(x)=2.5x^{2}+21x + 36).

Step1: Calculate (f(10))

Substitute (x = 10) into (f(x)): [ \begin{align*} f(10)&=2.5\times(10)^{2}+21\times10 + 36\ &=2.5\times100+210 + 36\ &=250+210+36\ &=496 \end{align*} ]

Step2: Calculate (f(5))

Substitute (x = 5) into (f(x)): [ \begin{align*} f(5)&=2.5\times(5)^{2}+21\times5+36\ &=2.5\times25 + 105+36\ &=62.5+105 + 36\ &=203.5 \end{align*} ]

Step3: Calculate the average rate of change

Use the formula (\frac{f(b)-f(a)}{b - a}), where (b = 10), (a = 5), (f(10)=496) and (f(5)=203.5) [ \begin{align*} \frac{f(10)-f(5)}{10 - 5}&=\frac{496-203.5}{5}\ &=\frac{292.5}{5}\ &=58.5 \end{align*} ]

The average rate of change of the function over the interval (5<x<10) is (58.5)

Explanation:

Step1: Evaluate (f(10))

Substitute (x = 10) into (f(x)) formula.

Step2: Evaluate (f(5))

Substitute (x = 5) into (f(x)) formula.

Step3: Apply average - rate - of - change formula

Use (\frac{f(10)-f(5)}{10 - 5}) to find result.