Consumers' Surplus The demand function for a certain brand of CD is given by
p = −0.01x2 − 0.3x + 27
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 (in dollars) if the market price is set at $9/disc. (Round your answer to two decimal places.)
$ 


To find the consumer's surplus, we'll follow the same steps as before: 

### Step 1: Find the quantity demanded $ x $ when the market price is $9 per disc.

Given the demand function:

\[
p(x) = -0.01x^2 - 0.3x + 27
\]

We set $ p(x) = 9 $ and solve for $ x $:

\[
9 = -0.01x^2 - 0.3x + 27
\]

Rearrange the equation:

\[
0.01x^2 + 0.3x + (9 - 27) = 0
\]

\[
0.01x^2 + 0.3x - 18 = 0
\]

Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where $ a = 0.01 $, $ b = 0.3 $, and $ c = -18 $:

\[
x = \frac{-0.3 \pm \sqrt{(0.3)^2 - 4(0.01)(-18)}}{2(0.01)}
\]

\[
x = \frac{-0.3 \pm \sqrt{0.09 + 0.72}}{0.02}
\]

\[
x = \frac{-0.3 \pm \sqrt{0.81}}{0.02}
\]

\[
x = \frac{-0.3 \pm 0.9}{0.02}
\]

This gives two possible solutions for $ x $:

\[
x_1 = \frac{0.6}{0.02} = 30 \quad \text{(positive solution, physically meaningful)}
\]
\[
x_2 = \frac{-1.2}{0.02} = -60 \quad \text{(negative, not meaningful in this context)}
\]

So, the quantity demanded when the price is $9 is $ x = 30 $ (which is 30,000 units).

### Step 2: Compute the consumer's surplus.

The consumer's surplus is the area under the demand curve from $ x = 0 $ to $ x = 30 $, minus the area of the rectangle formed by the market price $ p = 9 $ over the same interval:

\[
\text{Consumer's Surplus} = \int_{0}^{30} p(x) \, dx - (30 \times 9)
\]

First, compute the integral:

\[
\int_{0}^{30} \left(-0.01x^2 - 0.3x + 27\right) dx
\]

This integral is:

\[
\int \left(-0.01x^2 - 0.3x + 27\right) dx = -0.01 \frac{x^3}{3} - 0.3 \frac{x^2}{2} + 27x
\]

Evaluate from 0 to 30:

\[
\left[-0.01 \frac{30^3}{3} - 0.3 \frac{30^2}{2} + 27(30)\right] - \left[-0.01 \frac{0^3}{3} - 0.3 \frac{0^2}{2} + 27(0)\right]
\]

\[
= \left[-0.01 \frac{27000}{3} - 0.3 \frac{900}{2} + 810\right]
\]

\[
= \left[-90 - 135 + 810\right]
\]

\[
= 585
\]

Now, subtract the area of the rectangle:

\[
\text{Area of the rectangle} = 30 \times 9 = 270
\]

Thus, the consumer's surplus is:

\[
585 - 270 = 315
\]

So, the consumer's surplus is **$315.00**.

Would you like further details, or do you have any questions?

Here are some related questions you might find interesting:
1. How does consumer surplus change if the market price is $15 per disc?
2. What assumptions are made when calculating consumer surplus?
3. How is the concept of elasticity related to consumer surplus?
4. How would a change in the demand curve coefficients affect the consumer surplus?
5. Can you explain producer surplus in a similar context?
6. What impact does consumer surplus have on overall economic welfare?
7. How do taxes affect consumer surplus?
8. What is the difference between consumer surplus and total surplus?

**Tip:** Understanding how to set up the integral and correctly apply the limits is crucial in calculating consumer surplus. Double-check each step to ensure accuracy.

Learn how to calculate consumer's surplus for a CD demand function with a market price set at $9 per disc. Follow a step-by-step solution involving the quadratic formula and integral calculus. Understand the economic concept of consumer surplus and its application in pricing decisions.

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