Equation: Suppose that the demand of a certain item is x = -0.7 p + 20.
Evaluate the elasticity at E = 8.
I'm not sure what steps to take to solve this. I know that elasticity is equal to the absolute value of [(p/q) x (dp/dq)]
Well, let me take you on a delightful journey to find that elasticity value!
To start, we need to find the derivative of the demand equation with respect to price (dp/dq). So hold on tight, we're about to do some mathematical acrobatics!
The given equation is x = -0.7p + 20. We can rewrite it as p = (20 - x) / 0.7.
Now, let's differentiate with respect to p. The derivative of p with respect to p is 1, and the derivative of x with respect to p is -0.7. So dp/dx = -0.7.
Next, we need to calculate p/q. The quantity (q) is given as E = 8, so p/q = p/8.
Now, the elasticity formula tells us to multiply p/q by dp/dq. Since we found dp/dq to be -0.7, we can substitute these values into the formula.
Elasticity = |(p/q) * (dp/dq)|
= |(p/8) * (-0.7)|
And there you have it, the exciting journey through derivatives and substitutions leads us to the elasticity value. Just plug in the value of p, which we found to be (20-x)/0.7, and you'll have your answer! Happy calculating!
To evaluate the elasticity at a given point, you need to find the derivative of the demand equation with respect to price (dp/dq). Let's go through the steps to solve this problem:
Step 1: Given demand equation: x = -0.7p + 20
Step 2: Recognize that x represents quantity demanded, and p represents price.
Step 3: Rearrange the demand equation to get p in terms of x: p = (20 - x) / 0.7
Step 4: Differentiate the equation with respect to x: dp/dx = -1/0.7 = -10/7
Step 5: Identify E, which represents the price elasticity of demand: E = (p/x) * (dp/dx)
Step 6: Plug in the given values into the equation: E = (p/x) * (dp/dx) = (8) * (-10/7)
To find the numerical value of the elasticity, we substitute E = 8 into the equation and solve:
E = 8 * (-10/7) = -80/7
Therefore, the elasticity at E = 8 is approximately -11.43.
Note: Since elasticity is a ratio, it is always positive, so we take the absolute value of the result, which is 11.43, to obtain the final answer.