Answer;
The elasticity of demand can be found using the formula (dQ/dP) * (P/Q). By differentiating the demand function and substituting the given price, we can calculate the change in quantity. Then, by substituting the quantity demanded and price into the elasticity formula, we find the elasticity. In this case, the elasticity is 2/23197, indicating elastic demand.
Step-by-step explanation:
To find the elasticity of demand, we can use the formula: E = (dQ/dP) * (P/Q). First, we need to find the derivative of the demand function with respect to price (dQ/dP) and substitute the given price to get the change in quantity.
Then, we can substitute the quantity demanded and price into the equation to find the elasticity.
Given: q = 336/(4x+19)² and x = 1
Step 1: Find the derivative of the demand function:
dQ/dP = d(q)/d(x) * d(x)/d(P)
Step 2: Substituting the given price x = 1 into the demand function gives q = 336/(4(1)+19)² = 336/23² = 1.03
Step 3: Substitute the quantity demanded q = 1.03 and the given price x = 1 into the elasticity formula:
E = (dQ/dP) * (P/Q) = (dQ/dP) / (dP/dQ) = (dQ/dP) / (1/dQ/dP) = (dQ/dP)²
Step 4: Calculate the derivative:
dQ/dP = 8/(4x+19)³ = 8/(4(1)+19)^3 = 8/23^3 = 8/12167
Step 5: Substitute the derivative back into the elasticity equation:
E = (8/12167)² = 64/148154689 = 2/23197
Hence, the elasticity of demand at the given price is 2/23197, which makes the demand elastic.