Question

an appliance company determines that in order to sell $x$ dishwashers in a year, they should use a unit price of $p = 640 - 0.2x$ dollars. how many dishwashers must the company sell per year to maximize revenue?
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Explanation:

Step1: Define revenue function

Revenue $R(x)$ is price times quantity, so $R(x) = x \cdot p = x(640 - 0.2x) = 640x - 0.2x^2$

Step2: Find critical point

Take derivative of $R(x)$: $R'(x) = 640 - 0.4x$. Set to 0:
$640 - 0.4x = 0$

Step3: Solve for x

Rearrange to isolate $x$:
$0.4x = 640$
$x = \frac{640}{0.4} = 1600$

Step4: Confirm maximum

The second derivative $R''(x) = -0.4 < 0$, so this is a maximum.

Answer:

1600 dishwashers