Let's solve the given price-demand equation step by step.

The price-demand equation is:

$p + 0.002x = 50$

Part A: Express the demand $x$ as a function of the price $p$.

To find $x$ as a function of $p$, solve for $x$:

$p + 0.002x = 50$

Subtract $p$ from both sides:

$0.002x = 50 - p$

Now, divide by 0.002:

$x = \frac{50 - p}{0.002}$

Simplify the denominator:

$x = 50000 - 500p$

So, the demand function is:

$x(p) = 50000 - 500p$

Part B: Find the elasticity of demand, $E(p)$.

The elasticity of demand formula is given by:

$E(p) = \left( \frac{p}{x} \right) \cdot \frac{dx}{dp}$

First, we need to compute $\frac{dx}{dp}$. From the function $x(p) = 50000 - 500p$, the derivative of $x$ with respect to $p$ is:

$\frac{dx}{dp} = -500$

Now substitute $\frac{dx}{dp}$, $p$, and $x$ into the elasticity formula:

$E(p) = \left( \frac{p}{50000 - 500p} \right) \cdot (-500)$

Simplify the expression:

$E(p) = \frac{-500p}{50000 - 500p}$

Part C: Find the elasticity of demand when $p = 30$.

Substitute $p = 30$ into the elasticity formula:

$E(30) = \frac{-500(30)}{50000 - 500(30)}$

Simplify:

$E(30) = \frac{-15000}{50000 - 15000} = \frac{-15000}{35000} = -\frac{3}{7}$

So, the elasticity of demand when $p = 30$ is:

$E(30) = -\frac{3}{7}$

Interpretation of Elasticity and Revenue Change

The value $E(30) = -\frac{3}{7}$ is less than 1 in absolute value, meaning that the demand is inelastic at this price. When the demand is inelastic, increasing the price will increase total revenue.

If the price is increased by 10%, the revenue will:

Answer:
C. Revenue will increase.

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Would you like to see further details or explanations on any part of this?

Here are five follow-up questions for you:

1. How do you interpret the elasticity when $E(p) = 1$?
2. What happens to revenue when demand is elastic?
3. How would you calculate the new demand if the price increased by 10%?
4. What is the significance of the sign of elasticity?
5. How do we classify demand when $E(p) = 0$?

Tip: When elasticity is less than 1 in absolute value, demand is inelastic, meaning consumers are less sensitive to price changes.