The elasticity of demand is calculated using the formula:

$E(x) = -\frac{dD(x)}{dx} \cdot \frac{x}{D(x)}$

Here, $D(x) = 3900 - 30x$, so the steps are as follows:

Step 1: Differentiate $D(x)$ with respect to $x$

$\frac{dD(x)}{dx} = -30$

Step 2: Substitute $D(x)$ and $\frac{dD(x)}{dx}$ into the elasticity formula

$E(x) = -(-30) \cdot \frac{x}{3900 - 30x}$ $E(x) = 30 \cdot \frac{x}{3900 - 30x}$

Step 3: Solve for unit elasticity ($E(x) = 1$)

Set $E(x) = 1$ and solve for $x$: $1 = 30 \cdot \frac{x}{3900 - 30x}$

Multiply through by $3900 - 30x$ to eliminate the denominator: $3900 - 30x = 30x$

Simplify: $3900 = 60x$

Solve for $x$: $x = \frac{3900}{60} = 65$

Final Answer:

The price at which the elasticity of demand is 1 (unit elasticity) is 65 cents.

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Do you want further details or examples of similar calculations? Here are five related questions to consider:

1. How does elasticity affect total revenue for the company?
2. What happens to demand when the price is set above or below 65 cents?
3. Can you explain how to determine the price elasticity of demand in real-world scenarios?
4. What is the interpretation of elasticity values greater or less than 1?
5. How could this beverage company optimize pricing based on elasticity?

Tip: Always calculate elasticity at different price points to assess how sensitive demand is to price changes!