Studdy AI Logo
Math  /  Algebra

QuestionA bakery works out a demand function for its chocolate chip cookies and finds it to be q=D(x)=706−11xq=D(x)=706-11 x, where qq is the quantity of cookies sold when the price per cookie, in cer is xx. Use this information to answer parts a) through ff ). a) Find the elasticity E(x)=□E(x)=\square b) At what price is the elasticity of demand equal to 1 ? □\square f (Round to the nearest cent as needed.) c) At what prices is the elasticity of demand elastic? A. Prices cannot be elastic in this case B. Prices are elastic at all values. C. Greater than 32∅32 \varnothing D. Less than 32 c d) At what prices is the elasticity of demand inelastic?

Studdy Solution

STEP 1

1. The demand function is given by q=D(x)=706−11x q = D(x) = 706 - 11x .
2. Elasticity of demand, E(x) E(x) , is defined as E(x)=xq⋅dqdx E(x) = \frac{x}{q} \cdot \frac{dq}{dx} .
3. We need to find the elasticity function, the price at which elasticity equals 1, and determine the price ranges for elastic and inelastic demand.

STEP 2

1. Calculate the elasticity of demand function E(x) E(x) .
2. Determine the price at which elasticity equals 1.
3. Identify the price range where demand is elastic.
4. Identify the price range where demand is inelastic.

STEP 3

To find the elasticity, we first need to calculate the derivative of the demand function q=706−11x q = 706 - 11x .
The derivative dqdx \frac{dq}{dx} is: dqdx=−11 \frac{dq}{dx} = -11

STEP 4

Substitute dqdx=−11 \frac{dq}{dx} = -11 and q=706−11x q = 706 - 11x into the elasticity formula: E(x)=x706−11x⋅(−11) E(x) = \frac{x}{706 - 11x} \cdot (-11)
Simplify the expression: E(x)=−11x706−11x E(x) = \frac{-11x}{706 - 11x}

STEP 5

Set the elasticity equal to 1 to find the price: −11x706−11x=1 \frac{-11x}{706 - 11x} = 1
Solve for x x : −11x=706−11x -11x = 706 - 11x 0=706 0 = 706 x=32 x = 32

STEP 6

For demand to be elastic, E(x)>1 E(x) > 1 .
Solve: −11x706−11x>1 \frac{-11x}{706 - 11x} > 1
This inequality simplifies to: x<32 x < 32

STEP 7

For demand to be inelastic, E(x)<1 E(x) < 1 .
Solve: −11x706−11x<1 \frac{-11x}{706 - 11x} < 1
This inequality simplifies to: x>32 x > 32
The solutions are: a) E(x)=−11x706−11x E(x) = \frac{-11x}{706 - 11x} b) Price at which elasticity equals 1 is x=32 x = 32 cents. c) Prices are elastic at x<32 x < 32 cents. d) Prices are inelastic at x>32 x > 32 cents.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord