Question

the revenue (in dollars) from the sale of x car seats for infants is given by the following function. r(x)=24x - 0.010x^2 0≤x≤2400 (a) find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats. (b) use the four - step process to find r(x). (c) find the revenue and the instantaneous rate of change of revenue at a production level of 1,000 car seats, and interpret the results. (a) find the average change in revenue if production is changed from 1,000 car seats to 1,050 car seats. (round to one decimal place as needed.)

Explanation:

Step1: Recall average - rate - of - change formula

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, $f(x)=R(x)=24x - 0.010x^{2}$, $a = 1000$, and $b = 1050$.

Step2: Calculate $R(1000)$

$R(1000)=24\times1000-0.010\times(1000)^{2}=24000 - 10000=14000$.

Step3: Calculate $R(1050)$

$R(1050)=24\times1050-0.010\times(1050)^{2}=25200-0.010\times1102500=25200 - 11025=14175$.

Step4: Calculate the average rate of change

$\frac{R(1050)-R(1000)}{1050 - 1000}=\frac{14175 - 14000}{50}=\frac{175}{50}=3.5$.

Answer:

$3.5$