2.5 The price-demand equation for gasoline is
$$
0.2 x+5 p=80
$$
where $p$ is the price per gallon and $x$ is the daily demand measured in millions of gallons.
(a) Write the demand $f(p)$ as a function of price.
$$
f(p)=80-50 / 02=400-200
$$
(b) Sum What is the demand if the price is $\$ 4.00$ per gallon? Use the correct units to express your answer.
$$
\begin{aligned}
f(4) & =400-25(4) \
& =400-100 \
& =300 \text { millions gollors }
\end{aligned}
$$
(c) Sum Find an expression for revenue as a function of price.
$$
R(p)=(x)(p)=
$$
2.6 The price-demand equation for selling DVD players is
$$
x+2 p=600
$$
where $x$ is the demand (in thousands) for DVD players at a price of $p$ dollars. The graph is shown below. Label the tick marks on both axes.

Q: 2.5 The price-demand equation for gasoline is \[ 0.2 x+5 p=80 \] where $p$ is the price per gallon and $x$ is the daily demand measured in millions of gallons. (a) Write the demand $f(p)$ as a function of price. \[ f(p)=80-50 / 02=400-200 \] (b) Sum What is the demand if the price is $\$ 4.00$ per gallon? Use the correct units to express your answer. \[ \begin{aligned} f(4) & =400-25(4) \\ & =400-100 \\ & =300 \text { millions gollors } \end{aligned} \] (c) Sum Find an expression for revenue as a function of price. \[ R(p)=(x)(p)= \] 2.6 The price-demand equation for selling DVD players is \[ x+2 p=600 \] where $x$ is the demand (in thousands) for DVD players at a price of $p$ dollars. The graph is shown below. Label the tick marks on both axes.

∻Solution by Steps∻ ‖ step 1 ⋮ To express the demand $ f(p) $ as a function of price from the equation $ 0.2x + 5p = 80 $, we isolate $ x $: ‖ ‖ step 2 ⋮ Rearranging gives $ 0.2x = 80 - 5p $, thus $ x = \frac{80 - 5p}{0.2} $. Simplifying this results in $ x = 400 - 25p $. ‖ ‖ step 3 ⋮ Therefore, the demand function is $ f(p) = 400 - 25p $. ‖ ‖ step 4 ⋮ To find the demand when the price is $4.00 per gallon, substitute $ p = 4 $ into the demand function: ‖ ‖ step 5 ⋮ $ f(4) = 400 - 25(4) = 400 - 100 = 300 $. ‖ ‖ step 6 ⋮ Thus, the demand at a price of $4.00 per gallon is $ 300 $ million gallons. ‖ ‖ step 7 ⋮ To find the revenue function $ R(p) $, we use the formula $ R(p) = x \cdot p $. Substituting $ x = 400 - 25p $ gives: ‖ ‖ step 8 ⋮ $ R(p) = (400 - 25p)p = 400p - 25p^2 $. ‖ ‖ step 9 ⋮ Therefore, the revenue function is $ R(p) = 400p - 25p^2 $. ‖ ∻Answer∻ ⚹ (a) $ f(p) = 400 - 25p $, (b) Demand at $4.00 is $ 300 $ million gallons, (c) Revenue function $ R(p) = 400p - 25p^2 $ ⚹ ∻Key Concept∻ ⚹ Demand functions relate price to quantity demanded, while revenue functions relate price to total income from sales. ⚹ ∻Explanation∻ ⚹ The demand function shows how quantity demanded decreases as price increases, and the revenue function illustrates how total revenue changes with price. ⚹◊What is the relationship between price and demand in the equation $ x + 2p = 600 $ for DVD players?⍭ Generate me a similar question◊

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Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To express the demand $ f(p) $ as a function of price from the equation $ 0.2x + 5p = 80 $, we isolate $ x $: ‖ ‖ step 2 ⋮ Rearranging gives $ 0.2x = 80 - 5p $, thus $ x = \frac{80 - 5p}{0.2} $. Simplifying this results in $ x = 400 - 25p $. ‖ ‖ step 3 ⋮ Therefore, the demand function is $ f(p) = 400 - 25p $. ‖ ‖ step 4 ⋮ To find the demand when the price is $4.00 per gallon, substitute $ p = 4 $ into the demand function: ‖ ‖ step 5 ⋮ $ f(4) = 400 - 25(4) = 400 - 100 = 300 $. ‖ ‖ step 6 ⋮ Thus, the demand at a price of $4.00 per gallon is $ 300 $ million gallons. ‖ ‖ step 7 ⋮ To find the revenue function $ R(p) $, we use the formula $ R(p) = x \cdot p $. Substituting $ x = 400 - 25p $ gives: ‖ ‖ step 8 ⋮ $ R(p) = (400 - 25p)p = 400p - 25p^2 $. ‖ ‖ step 9 ⋮ Therefore, the revenue function is $ R(p) = 400p - 25p^2 $. ‖ ∻Answer∻ ⚹ (a) $ f(p) = 400 - 25p $, (b) Demand at $4.00 is $ 300 $ million gallons, (c) Revenue function $ R(p) = 400p - 25p^2 $ ⚹ ∻Key Concept∻ ⚹ Demand functions relate price to quantity demanded, while revenue functions relate price to total income from sales. ⚹ ∻Explanation∻ ⚹ The demand function shows how quantity demanded decreases as price increases, and the revenue function illustrates how total revenue changes with price. ⚹◊What is the relationship between price and demand in the equation $ x + 2p = 600 $ for DVD players?⍭ Generate me a similar question◊

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