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Market price is the current price at which a good or service can be bought or sold. In this exercise, the market price is provided as:

• \( \$32 \) per unit

To find out how many units of the commodity will be bought at this market price, we need to equate the demand function to the market price and solve for the quantity demanded:
\( D(q) = 32 \)

• \( 50 - 3q - q^2 = 32 \)

By solving this equation, students can determine the quantity demanded at this price.

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \). In our situation, the quadratic equation arises when we set the demand function equal to the market price:

• \( 50 - 3q - q^2 = 32 \)

Rearranging, we get:

• \( -q^2 - 3q + 18 = 0 \)

This standard form allows us to solve for \( q \) using the quadratic formula:

• \( q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = -1 \), \( b = -3 \), and \( c = 18 \).
By solving, we find the possible values for \( q \) and determine the valid (non-negative) solution.

Willingness to spend refers to the total amount of money consumers are ready to pay to purchase a specific number of units. It can be found by integrating the demand function from 0 to the quantity demanded. In this case, the quantity demanded when the market price is \( 32 \) is found to be 3 units. Let's integrate to find the willingness to spend:

• \( \int_{0}^{3} (50 - 3q - q^2) dq \)

By solving, we get the total area under the demand curve up to \( q = 3 \), representing the total willingness to spend.

An integral calculates the area under a curve in a given interval. This is particularly useful in finding consumers' surplus. Here, we calculate the area under the demand curve from 0 to the quantity demanded:

• \( \int_{0}^{3} (50 - 3q - q^2) dq \)

After solving the integral, we subtract the area that consumers actually pay, given by the product of the market price and the quantity demanded. This gives us the consumers' surplus:

• \( \text{Surplus} = \int_{0}^{3} (50 - 3q - q^2) dq - (32 \times 3) \)

By working out these calculations, we get to understand the additional benefit consumers receive from purchasing at a market price lower than they are willing to pay.