Points: 0 of 1 The price $$ p $$ (in dollars) and demand $$ x $$ for wireless headphones are related by $$ x=8,000-0.1 p^{2} $$. The current price of $$ \$ 110 $$ is decreasing at a rate $$ \$ 5 $$ per week. Find the associated revenue function $$ R(p) $$ and the rate of change (in dollars per week) of revenue. $$ \begin{array}{l} R(p)=8000 p-0.1 p^{3} \ \frac{d}{d t}(R(p))=\square \end{array} $$ $$ \square $$ $$ \begin{array}{r} 8,000-0.3 p^{2} \frac{d p}{d t} \ 8,000-0.3 p^{2} \ 8,000 p-0.1 p^{3} \frac{d p}{d t} \ \frac{d p}{d t}\left(8,000 p-0.1 p^{3}\right) \ 8,000 p \frac{d p}{d t}-0.1 p^{3} \frac{d p}{d t} \ 8,000 \frac{d p}{4}-0.3 p^{2} \frac{d p}{\prime} \end{array} $$

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UpStudy AI · Accepted Answer

We start with the demand function

$$
x=8000-0.1p^2.
$$

Since revenue is the product of price and quantity sold, we have

$$
R(p)=p\cdot x=p\left(8000-0.1p^2\right)=8000p-0.1p^3.
$$

To find the rate of change of revenue with respect to time, we apply the chain rule:

$$
\frac{dR}{dt}=\frac{dR}{dp}\cdot\frac{dp}{dt}.
$$

Differentiate $$R(p)=8000p-0.1p^3$$ with respect to $$p$$:

$$
\frac{dR}{dp}=8000-0.3p^2.
$$

Thus,

$$
\frac{dR}{dt}=\left(8000-0.3p^2\right)\frac{dp}{dt}.
$$

At the current price $$p=110$$ dollars and given that the price is decreasing at a rate $$ \frac{dp}{dt}=-5$$ dollars per week, substitute these values into the expression:

$$
\frac{dR}{dt}=\left(8000-0.3(110)^2\right)(-5).
$$

Calculate $$110^2$$:

$$
110^2=12100.
$$

Multiply by 0.3:

$$
0.3\times 12100=3630.
$$

Now substitute back:

$$
\frac{dR}{dt}=\left(8000-3630\right)(-5)=4370\times(-5)=-21850.
$$

Thus, the revenue is decreasing at a rate of

$$
\boxed{-21850\text{ dollars per week}}.
$$
The revenue is decreasing at a rate of \$21,850 per week.

Points: 0 of 1

The price (in dollars) and demand for wireless headphones are related by . The current price of is decreasing at a rate per week. Find the associated revenue function and the rate of change (in dollars per week) of revenue.

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Points: 0 of 1 The price (in dollars) and demand for wireless headphones are related by . The current price of is decreasing at a rate per week. Find the associated revenue function and the rate of change (in dollars per week) of revenue.