Math Problem Statement

Use the price-demand equation to find E (p), the elasticity of demand x=f(p)=155-30 in (p)

Solution

To find the elasticity of demand E(p)E(p), we need to apply the formula for elasticity in terms of price-demand function:

E(p)=pxdxdpE(p) = \frac{p}{x} \cdot \frac{dx}{dp}

Given: x=f(p)=15530ln(p)x = f(p) = 155 - 30 \ln(p)

Let's proceed step-by-step:

Step 1: Find dxdp\frac{dx}{dp}

Differentiate x=15530ln(p)x = 155 - 30 \ln(p) with respect to pp:

dxdp=30p\frac{dx}{dp} = -\frac{30}{p}

Step 2: Substitute into the elasticity formula

Now we substitute x=15530ln(p)x = 155 - 30 \ln(p) and dxdp=30p\frac{dx}{dp} = -\frac{30}{p} into the elasticity formula:

E(p)=p15530ln(p)(30p)E(p) = \frac{p}{155 - 30 \ln(p)} \cdot \left(-\frac{30}{p}\right)

Step 3: Simplify the expression

The pp terms cancel out, giving:

E(p)=3015530ln(p)E(p) = \frac{-30}{155 - 30 \ln(p)}

Thus, the elasticity of demand E(p)E(p) is:

E(p)=3015530ln(p)E(p) = \frac{-30}{155 - 30 \ln(p)}

Would you like further details on this process or have any other questions?


Here are some related questions you might consider:

  1. What is the interpretation of a negative elasticity value?
  2. How does the elasticity of demand change as pp increases?
  3. What does it mean if E(p)=1E(p) = -1 at a particular price pp?
  4. How can this elasticity value be used to predict revenue changes with price adjustments?
  5. What are the conditions under which this demand function would be considered elastic or inelastic?

Tip: Remember, elasticity greater than 1 (in magnitude) indicates elastic demand, less than 1 indicates inelastic demand, and exactly 1 means unitary elasticity.

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Math Problem Analysis

Mathematical Concepts

Elasticity of Demand
Differentiation
Price-Demand Functions

Formulas

E(p) = (p / x) * (dx/dp)
x = f(p) = 155 - 30ln(p)
dx/dp = -30/p

Theorems

Elasticity formula
Chain Rule in Differentiation

Suitable Grade Level

Grades 11-12