A company is planning to manufacture a new blender. After conducting extensive market surveys, the research department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per blender. Assuming the demand equation is linear, use the research department's estimates to find the revenue equation in terms of the demand x.
A) R(x) = 20x + $\frac { x ^ { 2 } } { 20 }$
B) R(x) = 80x - $\frac { x ^ { 2 } } { 20 }$
C) R(x) = 80x - 20 $x ^ { 2 }$
D) R(x) = 80x - 20

Question

Quizplus · Accepted Answer

The revenue equation, R(x), can be found by first determining the price-demand equation, which is linear according to the given information. Using the two points (600, 50) and (800, 40), we can find the slope of the line: $m = \frac{40 - 50}{800 - 600} = \frac{-10}{200} = -\frac{1}{20}$. The price-demand equation is of the form $p = mx + b$, where $p$ is the price, $x$ is the quantity demanded, and $b$ is the y-intercept. Substituting one of the points into the equation along with the slope, we can solve for $b$: $50 = -\frac{1}{20}(600) + b$, leading to $b = 80$. Thus, the price-demand equation is $p = 80 - \frac{1}{20}x$. The revenue equation is $R(x) = x \cdot p = x(80 - \frac{1}{20}x) = 80x - \frac{x^2}{20}$, which matches choice B.

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