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Lunna [17]
3 years ago
14

The John Deere company has found that revenue, in dollars, from sales of heavy-duty tractors is a function of the unit price p,

in dollars, that it charges. If the revenue R is R(p)=-1/2 p^2+1900p, please answer the following questions:
Part A: How much revenue will the company generate if they sell 700 tractors?

Part B: How many tractors does the company want to sell to maximize the revenue?

Show your work/explain your answers please!

Mathematics
1 answer:
lys-0071 [83]3 years ago
8 0

<u>Answer-</u>

<em>A- 700 tractors will generate a revenue of $1,085,000</em>

<em>B- The company must sell 1900 tractors in order to maximize the revenue.</em>

<u>Solution-</u>

The John Deere company has found that the revenue from sales of heavy-duty tractors is a function of the unit price p, in dollars, that it charges. If the revenue R, in dollars, is

R(p)=-\frac{1}{2}p^2+1900p

<u>How much revenue will the company generate if they sell 700 tractors:</u>

\Rightarrow R(p)=-\frac{1}{2}(700)^2+1900(700)

\Rightarrow R(p)=1,085,000

Therefore, 700 tractors will generate a revenue of $1,085,000

<u>How many tractors does the company want to sell to maximize the revenue</u>

By calculating the value of p for which R(p) is maximum will be the number of tractors for which the revenue will be maximum. We can take the help of derivatives to find this.

R(p)=-\frac{1}{2}p^2+1900p

Taking the derivative w.r.t p,

\Rightarrow R' (p)=-\frac{1}{2}\times 2\times p+1900

\Rightarrow R' (p)=-p+1900

\Rightarrow R' (p)=1900-p

Taking R'(p) = 0, to find out the critical points

\Rightarrow R' (p)=0

\Rightarrow 1900-p=0

\Rightarrow p=1900

So, at p = 1900, the function R(p) will be maximum. The maximum value will be,

\Rightarrow R(p)=-\frac{1}{2}(1900)^2+1900(1900)

\Rightarrow R(p)=1,805,000

Therefore, the company must sell 1900 tractors in order to maximize the revenue and the maximum revenue will be $1,805,000.


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