Final answer:
The elasticity of demand is calculated by determining the percentage change in quantity demanded over the percentage change in price, which requires differentiation and substitution into the elasticity formula. One must solve for x in the given price-demand equation, differentiate it with respect to price, and then apply the elasticity formula to obtain the correct relationship between elasticity and price.
Step-by-step explanation:
To find the formula for the elasticity of demand at price p, based on the price-demand equation 40p + x = 5,000, we need to understand the concept of elasticity first. Elasticity of demand is computed as the percentage change in quantity demanded divided by the percentage change in price. According to the information provided, we need to determine how the quantity demanded changes with the price, which is given by x in this equation.
First, we solve for x to make it the subject: x = 5,000 - 40p. To find the elasticity, we then need to differentiate x with respect to p, which gives us dx/dp = -40. The elasticity formula is the absolute value of (dx/dp) times (p/x).
Now, substitute x = 5,000 - 40p into the elasticity equation to get: E(p) = |dx/dp| * (p / (5,000 - 40p)) = |-40| * (p / (5,000 - 40p)) = 40 * (p / (5,000 - 40p)). After simplification, you'll arrive at the correct elasticity formula which best fits the equation given.