(1 point) A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the denand in a partlculer elly is gtren approximutety by

where thousand lipsticks were soid per week at a price of dollars each. At whut price will the weekly revenue be maximized?

Price = $
Note: the answer must an actual value for money, like 719.

Ask by Grant Nguyen. in the United States

Mar 25,2025

Question

(1 point) A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the denand in a partlculer elly is gtren approximutety by

where thousand lipsticks were soid per week at a price of dollars each. At whut price will the weekly revenue be maximized?

Price = $
Note: the answer must an actual value for money, like 719.

Ask by Grant Nguyen. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

Let $$ x $$ be the number (in thousands) of lipsticks sold per week. The price-demand relationship is given by
$$
p = \frac{8}{e^x}, \quad 0 \leq x \leq 2.
$$
Since the revenue $$ R $$ is the product of the number sold and the price, we have
$$
R(x) = 1000x \cdot \frac{8}{e^x} = \frac{8000x}{e^x}.
$$
Because the constant $$ 8000 $$ does not affect the location of the maximum, we can instead maximize
$$
f(x) = \frac{x}{e^x}.
$$

### Step 1. Differentiate $$ f(x) $$
We differentiate using the quotient rule (or by rewriting as $$ x e^{-x} $$):

$$
f(x) = x e^{-x}.
$$

Differentiate with respect to $$ x $$:
$$
f'(x) = e^{-x} + x(-e^{-x}) = e^{-x}(1 - x).
$$

### Step 2. Find the Critical Points
Set $$ f'(x) = 0 $$:
$$
e^{-x}(1 - x) = 0.
$$
Since $$ e^{-x} > 0 $$ for all $$ x $$, we have
$$
1 - x = 0 \quad \Longrightarrow \quad x = 1.
$$

### Step 3. Determine the Price at $$ x = 1 $$
At $$ x = 1 $$ (which corresponds to 1,000 lipsticks sold), the price $$ p $$ is:
$$
p = \frac{8}{e^1} = \frac{8}{e}.
$$
Numerically, using $$ e \approx 2.718 $$:
$$
p \approx \frac{8}{2.718} \approx 2.942.
$$

### Final Answer
The weekly revenue is maximized when the lipstick is sold at approximately 
$$
\boxed{\$2.94}.
$$
The price that maximizes weekly revenue is approximately \$2.94.

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