A company manufactures and sells $$ x $$ cellphones per week. The weekly price-demand and cost equations are given below. (A) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly revenue? What is the maximum weekly revenue? The company should produce 600 phones each week at a price of $$ \$ 300 $$. $$ \mathrm{C}(\mathrm{x})=20,000+130 \mathrm{x} $$. (Round to the nearest cent as needed.) The maximum weekly revenue is $$ \$ 180000 $$. (Round to the nearest cent as needed.) (B) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly profit? What is the maximum weekly profit? The company should produce $$ \square $$ phones each week at a price of $$ \$ \square $$. (Round to the nearest cent as needed.)

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UpStudy AI · Accepted Answer

We are given that the weekly price–demand function (inferred from part (A)) is  
$$
p(x)=600-0.5x,
$$
so the revenue function is  
$$
R(x)=x\cdot p(x)=600x-0.5x^2.
$$

The cost function is  
$$
C(x)=20000+130x.
$$

The profit function is revenue minus cost:  
$$
P(x)=R(x)-C(x)=\Bigl(600x-0.5x^2\Bigr)-\Bigl(20000+130x\Bigr).
$$
Simplify $$P(x)$$:  
$$
P(x)=-0.5x^2 + (600-130)x - 20000 = -0.5x^2+470x-20000.
$$

To maximize the profit function, we differentiate $$P(x)$$ with respect to $$x$$ and set the derivative equal to zero. That is,  
$$
P'(x)=-x+470.
$$
Set $$P'(x)=0$$:  
$$
-x+470=0 \quad\Longrightarrow\quad x=470.
$$

Thus, the profit is maximized when the company produces $$470$$ cellphones per week.

Now, we find the corresponding price by substituting $$x=470$$ into the price–demand function:  
$$
p(470)=600-0.5\cdot470=600-235=365.
$$
So the price should be $$\$365.00$$ per cellphone.

Next, we compute the maximum weekly profit by evaluating $$P(x)$$ at $$x=470$$:

$$
P(470)=-0.5(470)^2+470(470)-20000.
$$

First, calculate:
- $$470^2=220900$$,
- $$-0.5(220900)=-110450$$, 
- $$470(470)=220900$$.

Now combine:  
$$
P(470)=-110450+220900-20000.
$$
$$
P(470)= (220900-110450)-20000 = 110450-20000=90450.
$$

Therefore, the maximum weekly profit is $$\$90450.00$$.

In summary:

- The company should produce $$470$$ cellphones each week.
- The price to charge is $$\$365.00$$ per cellphone.
- The maximum weekly profit is $$\$90450.00$$.
The company should produce 470 cellphones each week at a price of \$365.00. The maximum weekly profit is \$90,450.00.

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