Questions: The quantity of a certain item sold when the price is p dollars is given by x=160-0.04 p. Remembering that revenue is given by R=x p, find the formula for revenue in terms of just p. Your answer should simplify into an equation of the form R=a p^2+b p+c. Write the values of a, b, and c below.

The quantity of a certain item sold when the price is p dollars is given by x=160-0.04 p. Remembering that revenue is given by R=x p, find the formula for revenue in terms of just p. Your answer should simplify into an equation of the form R=a p^2+b p+c. Write the values of a, b, and c below.
Transcript text: The quantity of a certain item sold when the price is $p$ dollars is given by $x=160-0.04 p$. Remembering that revenue is given by $R=x p$, find the formula for revenue in terms of just $p$. Your answer should simplify into an equation of the form $R=a p^{2}+b p+c$. Write the values of $a, b$, and $c$ below.
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Solution

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Solution Steps

Step 1: Substitute the given quantity equation into the revenue formula

The revenue formula is given by \( R = x p \). Substitute \( x = 160 - 0.04 p \) into this formula: \[ R = (160 - 0.04 p) p \]

Step 2: Expand the equation

Multiply \( p \) through the parentheses: \[ R = 160 p - 0.04 p^2 \]

Step 3: Rearrange the equation into the standard quadratic form

Rearrange the terms to match the form \( R = a p^2 + b p + c \): \[ R = -0.04 p^2 + 160 p \] Here, \( a = -0.04 \), \( b = 160 \), and \( c = 0 \).

Final Answer

The values are \( a = -0.04 \), \( b = 160 \), and \( c = 0 \).

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