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The shutdown point for a firm is the price level at which the firm can no longer cover its average variable costs and decides to halt production. In the given exercise, the shutdown point is determined by analyzing the supply function, which is 0 for prices less than 10. This implies that if the price drops below $10 per unit, producers are unable to sustain operations economically. Therefore, the shutdown point is at - When price, \( p = 10 \) dollars, the supply becomes positive.- Below \( p = 10 \) dollars, producing is economically unfeasible.This point is crucial for businesses as it marks the boundary between continuing production and suspending operations due to inadequate product pricing.

A demand function presents the relationship between the price of a commodity and the quantity demanded by consumers. The exercise provides a demand function: \( D(p) = 50 - 2p \), expressed in hundreds of units. This linear equation helps us understand how demand decreases as price increases.- \( D(p) \) is the quantity demanded at price \( p \).- For a price \( p \), \( D(10) = 50 - 20 = 30 \) hundred units shows that higher prices lead to lower demand.The demand function's slope, which in this case is \(-2\), graphically shows a downward slope indicating that as prices rise, demand falls. Understanding the demand function allows businesses to project how price changes might affect consumer purchasing behavior.

The supply function indicates how many units of goods producers are willing to supply at various price levels. For this exercise, the supply function given is piecewise:- For \( p < 10 \): \( S(p) = 0 \)- For \( p \geq 10 \): \( S(p) = 0.1p^2 \)In the context of the supply function:- No supply is available below a certain price threshold, \( p < 10 \), which signals an unsustainable market condition for sellers.- As the price becomes \( p \geq 10 \), the supply becomes a quadratic function \( 0.1p^2 \), showing how supply increases as price rises.This nonlinear expression reflects that supply generally increases with price, highlighting a typical supplier response to market signals.

The quadratic formula is a mathematical tool used to find the roots of quadratic equations of the form \( ax^2 + bx + c = 0 \). In this exercise, it was used to solve the equation derived from setting the demand equal to the supply: \( 50 - 2p = 0.1p^2 \).- Rearranging into the quadratic form gives \( 0.1p^2 + 2p - 50 = 0 \), then adjusted to \( p^2 + 20p - 500 = 0 \) to ease calculations by removing decimals.- The quadratic formula is given by \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).In this scenario:- \( a = 1 \), \( b = 20 \), \( c = -500 \)- The discriminant is \( 2400 \), and solving gives \( p = 10 \), ignoring the negative outcome.This illustrates how the quadratic formula aids in finding equilibrium price points where the market supply equals demand.