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

Answer:

To find the average rate of change of the function (y = f(x)=2.5x^{2}+21x + 36) over the interval (5\lt x\lt10), we use the formula (\frac{f(b)-f(a)}{b - a}), where (a = 5) and (b=10).

First, find (f(5)): [ \begin{align*} f(5)&=2.5\times5^{2}+21\times5 + 36\ &=2.5\times25+105 + 36\ &=62.5+105+36\ &=203.5 \end{align*} ]

Next, find (f(10)): [ \begin{align*} f(10)&=2.5\times10^{2}+21\times10+36\ &=2.5\times100 + 210+36\ &=250+210+36\ &=496 \end{align*} ]

Then, calculate the average rate of change: [ \begin{align*} \frac{f(10)-f(5)}{10 - 5}&=\frac{496-203.5}{5}\ &=\frac{292.5}{5}\ & = 58.5 \end{align*} ]

So the average rate of change of the function over the interval (5\lt x\lt10) is (58.5).

Explanation:

Step1: Recall average - rate - of - change formula

(\text{Average rate of change}=\frac{f(b)-f(a)}{b - a}), (a = 5), (b = 10)

Step2: Calculate (f(5))

[f(5)=2.5\times5^{2}+21\times5 + 36=203.5]

Step3: Calculate (f(10))

[f(10)=2.5\times10^{2}+21\times10+36=496]

Step4: Compute average rate of change

(\frac{f(10)-f(5)}{10 - 5}=\frac{496 - 203.5}{5}=58.5)