Given the demand function D ( p ) = √ 225 − 2 p , Find the Elasticity of Demand at a price of $65

Question

Can you explain how to solve this

Answer 1

D(p) = 15 - 2p

dD/dp = -2

dD = -2 dp

so for any marginal change in price , dp, the demand drops 2

Now if you meant D(p) =sqrt ( 225-2p)

= (225 - 2p)^(1/2) , where the entire binomial is under the square root, then

dD/dp = -2 /(225 - 2p)^1/2

in that case, -2/(225 - 2*65)^1/2 =

-2/(225 - 130)^1/2 =

-2/ (95)^1/2 =

-0.20519567041703081909133501473473

quite a different answer!

Answer 2

Elasticity of demand is the derivative of the demand function. To summarize the meaning of a derivative in this context, we're looking for the marginal change in demanded units for a marginal change in price, particularly around the point p=65.

Assuming the formula is D(p)=sqrt(225-2p), we calculate

dD/dp = d/dp( sqrt(225-2p) ) = -1/sqrt(225-2p)

Here's a refresher on computing this derivative: https://www.wikihow.com/Differentiate-the-Square-Root-of-X

With the derivative calculated, all that's left is to plug in p=65.

D'(p) = -1/sqrt(225-p)

D'(65) = -1/sqrt(160)

As far as that negative result goes: https://opentextbc.ca/principlesofeconomics/chapter/5-1-price-elasticity-of-demand-and-price-elasticity-of-supply/

Price elasticities of demand are always negative since price and quantity demanded always move in opposite directions (on the demand curve). By convention, we always talk about elasticities as positive numbers.