The demand function for a certain item is x = 35(sqrt(32-p^2))
---Evaluate the elasticity at 5. E(5)=
I don't know how to find elasticity, so i don't know how to start this problem. If someone could explain elasticity i think that i could figure it out.
Isn'n
Elasticity = - ddemand/dprice * P/q ?
so here
elasticity= - dx/dp * p/x
I assume you can take the derivative, put p=5 (and x(5)) and compute E.
dx/dp = 35*(1/2)*(-2p)/sqrt(32-p^2)
= -35/sqrt(32-p^2)
Now evaluate E(x) = -(p/x)* (dx/dp)
@ p=5
x(5) = 35 sqrt 5
and put it all together.
A few numbers will cancel, simplifying the result.
Elasticity measures how sensitive the demand or supply of a product is to changes in its price. In the case of the demand elasticity, it determines how much the quantity demanded of a product will change in response to a change in its price.
The formula for elasticity is:
E(p) = (dQ/dp) * (p/Q)
Where:
- E(p) refers to the elasticity at a given price value (p),
- dQ/dp represents the derivative of the demand function with respect to price (p),
- Q represents the quantity demanded at a particular price (p).
To evaluate elasticity at 5, we need to compute the derivative of the demand function with respect to price and substitute the value 5 into the equation.
Step 1: Take the derivative of the demand function with respect to price.
To differentiate the function, we keep the constant 35 and find the derivative of the inner function (sqrt(32-p^2)). Using the chain rule, we have:
dQ/dp = 35 * d(sqrt(32-p^2))/dp
Step 2: Simplify and replace with the given value of p.
Since we know p = 5, substitute it into the derivative expression. The value of p in the following steps is 5.
dQ/dp = 35 * d(sqrt(32-25))/dp
Step 3: Evaluate the derivative of sqrt(32-25).
To calculate the derivative, we need to express the square root function as a power. We have:
sqrt(32-25) = (32-25)^(1/2) = 7^(1/2) = sqrt(7)
Step 4: Substitute the evaluated derivative and the value of p.
dQ/dp = 35 * d(sqrt(7))/dp
Step 5: Calculate d(sqrt(7))/dp.
The derivative of sqrt(7) is zero since it is a constant value.
Step 6: Calculate the elasticity.
E(5) = (dQ/dp) * (p/Q) = 35 * 0 * (5 / (35*sqrt(32-25)))
Step 7: Simplify the expression.
E(5) = 0
Therefore, the elasticity at a price of 5 is 0. Please note that this result implies that the demand for the item is perfectly inelastic at that price point, meaning that a change in price will not affect the quantity demanded.