Answer the following questions for the​ price-demand equation.
p plus 0.002 x equals 50
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Part 1
​(A) Express the demand x as a function of the price p.
xequals
  
enter your response here
​(B) Find the elasticity of​ demand, Upper E left parenthesis p right parenthesis .
​E(p)equals
  
enter your response here
​(C) What is the elasticity of demand when pequals30​?
The elasticity of demand when pequals30 is 
  
enter your response here.
​(Type an integer or a simplified​ fraction.)
If the price is increased by 10 %​, how will revenue​ change?
A.
Revenue will decrease
B.
Revenue will remain the same
C.
Revenue will increase

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.**

---

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.

Answer the following questions for the price-demand equation: p + 0.002x = 50. (A) Express the demand x as a function of the price p. (B) Find the elasticity of demand, E(p). (C) What is the elasticity of demand when p = 30? If the price is increased by 10%, how will revenue change?

Discover how to express the demand function x(p), calculate the elasticity of demand E(p), and analyze revenue changes with price-demand equations. This problem uses algebra and calculus concepts, making it ideal for Grades 10-12 or college-level students.

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