# The demand function for a certain brand of CD is given by

p = −0.01x^2 − 0.2x + 10

where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the market price is set at $2/disc. (Round your answer to two decimal places.)

## Well, let's find the quantity demanded at a price of $2/disc by setting the demand function equal to 2.

-0.01x^2 - 0.2x + 10 = 2

Simplifying, we get:

-0.01x^2 - 0.2x + 8 = 0

Now we solve for x using the quadratic formula,

x = (-b ± √(b^2 - 4ac)) / 2a

where a = -0.01, b = -0.2, and c = 8.

Plugging in the values, we have:

x = (-(-0.2) ± √((-0.2)^2 - 4(-0.01)(8))) / (2(-0.01))

Simplifying:

x = (0.2 ± √(0.04 + 0.32)) / (-0.02)

x = (0.2 ± √0.36) / (-0.02)

x = (0.2 ± 0.6) / (-0.02)

x = -0.4 / (-0.02) or 0.8 / (-0.02)

x = 20 or -40

Since we can't have a negative quantity, we discard the -40 value.

Therefore, the quantity demanded at a price of $2/disc is 20 thousand units.

Now, let's calculate the consumers' surplus. Consumers' surplus can be found by calculating the area between the demand curve and the price line up to the quantity demanded.

The area of the consumers' surplus triangle can be calculated using the formula:

Consumers' surplus = 0.5 * (Quantity Demanded) * (Price - 0)

Plugging in the values:

Consumers' surplus = 0.5 * (20) * (2 - 0)

Consumers' surplus = 0.5 * 20 * 2

Consumers' surplus = 0.5 * 40

Consumers' surplus = 20

So, the consumers' surplus is $20.

## To determine the consumer's surplus, we need to integrate the demand function from zero to the quantity demanded at the market price.

Given:

Demand function: p = -0.01x^2 - 0.2x + 10

Market price: $2/disc

First, let's find the quantity demanded at the market price:

2 = -0.01x^2 - 0.2x + 10

Rearranging the equation and setting it equal to zero:

-0.01x^2 - 0.2x + 8 = 0

Solving this quadratic equation, we get two solutions: x ≈ 28.6 and x ≈ -1.4. Since the quantity demanded cannot be negative, we discard the negative solution.

So, the quantity demanded at the market price is approximately 28.6 thousand units.

Now, let's calculate the consumer's surplus by integrating the demand function from 0 to 28.6:

CS = ∫[0, 28.6] (-0.01x^2 - 0.2x + 10) dx

CS = ∫[-0.01x^2 - 0.2x + 10] dx evaluated from 0 to 28.6

To calculate the definite integral, we need to find the antiderivative of the integrand and evaluate it at the upper and lower limits:

CS = [(-0.01/3)x^3 - (0.2/2)x^2 + 10x] evaluated from 0 to 28.6

CS = (-0.01/3)(28.6^3) - (0.2/2)(28.6^2) + 10(28.6) - [(-0.01/3)(0^3) - (0.2/2)(0^2) + 10(0)]

Calculating this expression, we get:

CS ≈ 419.36

Therefore, the consumer's surplus is approximately $419.36.

## To determine the consumers' surplus, we need to find the difference between the maximum amount consumers are willing to pay for a CD and the market price.

1. Start by finding the maximum quantity demanded. To do this, we take the derivative of the demand function with respect to x and set it equal to zero.

p = -0.01x^2 - 0.2x + 10

Take the derivative:

dp/dx = -0.02x - 0.2

Set it equal to zero and solve for x:

-0.02x - 0.2 = 0

-0.02x = 0.2

x = 10

So, the maximum quantity demanded is 10,000 units.

2. Next, substitute this value into the demand function to find the maximum price consumers are willing to pay:

p = -0.01x^2 - 0.2x + 10

p = -0.01(10^2) - 0.2(10) + 10

p = -0.01(100) - 2 + 10

p = -1 - 2 + 10

p = 7

Therefore, consumers are willing to pay a maximum price of $7 per CD.

3. Finally, calculate the consumers' surplus by subtracting the market price from the maximum price:

Consumers' surplus = (Maximum price - Market price) * Quantity

Consumers' surplus = ($7 - $2) * 10,000

Consumers' surplus = $5 * 10,000

Consumers' surplus = $50,000

Therefore, the consumers' surplus is $50,000.

## 2 = −0.01x^2 − 0.2x + 10

200 = -x^2 -20x + 100

x^2 +20x -1000+ 200 = 0

x^2 + 20x -800 =0

(x -20) (x +40) =0

x = 20 using

x = -40 not feasible

ʃ (−0.01x^2 − 0.1x + 10)dx on [0,60] - 20*2

-1x^3/300 -1x^2/10 +10x

-8000/3-400/10+200-40 = 93.33