Question

The demand function for a certain brand of CD is given byp = −0.01x2 − 0.2x + 9where 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 $6/disc. (Round your answer to two decimal places.)

Answer 1

Sure, let's solve this step by step.

Firstly, let's find out the quantity demanded each week when the price is set at $6/disc. We can do this by setting p (price per disc) equal to 6 in our demand function. The equation becomes:

6 = -0.01x² - 0.2x + 9

Solving this quadratic equation for x (quantity demanded), we get two solutions: x = -30 and x = 10.

As quantity cannot be negative, we discard the solution x = -30 and keep x = 10. This means that when the price is $6/disc, the quantity demanded each week is 10,000 discs (remember that x is in thousands).

Now, let's calculate the consumer surplus. Consumer surplus is defined as the area under the demand curve and above the market price, from zero to the quantity demanded. Mathematically, it's defined by the integral of (p - 6) as x ranges from 0 to the quantity demanded.

In our case, the integral becomes:

∫(0 to 10) (-0.01x² - 0.2x + 9 - 6) dx

Solving this integral, we get the consumer surplus to be 16.67 (rounded to two decimal places).

So, the consumer surplus if the market price is set at $6/disc is $16.67 thousand.