Question

the revenue (in dollars) from the sale of x car seats for infants is given by the following function. r(x)=24x - 0.010x² 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. 3.5 (round to one decimal place as needed.) (b) r(x)=□

Explanation:

Step1: Recall the formula for average rate of change

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 $b = 1050$.
First, find $R(1000)$ and $R(1050)$.
$R(1000)=24\times1000-0.010\times(1000)^{2}=24000 - 10000=14000$.
$R(1050)=24\times1050-0.010\times(1050)^{2}=25200-0.010\times1102500=25200 - 11025=14175$.
Then, the average rate of change is $\frac{R(1050)-R(1000)}{1050 - 1000}=\frac{14175 - 14000}{50}=\frac{175}{50}=3.5$.

Step2: Use the four - step process to find $R^{\prime}(x)$

Step 1: Find $R(x + h)$
$R(x + h)=24(x + h)-0.010(x + h)^{2}=24x+24h-0.010(x^{2}+2xh + h^{2})=24x+24h-0.010x^{2}-0.020xh-0.010h^{2}$.
Step 2: Find $R(x + h)-R(x)$
$R(x + h)-R(x)=(24x+24h-0.010x^{2}-0.020xh - 0.010h^{2})-(24x - 0.010x^{2})=24h-0.020xh-0.010h^{2}$.
Step 3: Find $\frac{R(x + h)-R(x)}{h}$
$\frac{R(x + h)-R(x)}{h}=\frac{24h-0.020xh-0.010h^{2}}{h}=24 - 0.020x-0.010h$.
**Step 4: Find $\lim_{h
ightarrow0}\frac{R(x + h)-R(x)}{h}$**
$\lim_{h
ightarrow0}\frac{R(x + h)-R(x)}{h}=\lim_{h
ightarrow0}(24 - 0.020x-0.010h)=24 - 0.020x$. So, $R^{\prime}(x)=24 - 0.020x$.

Step3: Find $R(1000)$ and $R^{\prime}(1000)$

We already found $R(1000) = 14000$.
$R^{\prime}(1000)=24-0.020\times1000=24 - 20 = 4$.
Interpretation: The revenue from selling 1000 car - seats is $14000$ dollars. The instantaneous rate of change of revenue at a production level of 1000 car - seats is $4$ dollars per car - seat. This means that at a production level of 1000 car - seats, for each additional car - seat produced and sold, the revenue will increase by approximately $4$ dollars.

Answer:

(A) 3.5
(B) $R^{\prime}(x)=24 - 0.020x$
(C) $R(1000)=14000$, $R^{\prime}(1000)=4$. The revenue from selling 1000 car - seats is $14000$ dollars and the instantaneous rate of change of revenue at a production level of 1000 car - seats is $4$ dollars per car - seat, meaning for each additional car - seat produced and sold at this level, the revenue will increase by approximately $4$ dollars.