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The demand equation represents a mathematical representation of the relationship between the price of a good and the quantity demanded by consumers. In this context, the demand equation is typically expressed as a function, like in the problem: \[ p = 100 - 3\sqrt{q} \]In this equation, \(p\) stands for the price of the good, and \(q\) represents the quantity demanded. Here, the formula captures the price consumers are willing to pay based on the amount of the product they wish to purchase. The term \(3\sqrt{q}\) indicates that the price is adjusted by the square root of the quantity, multiplied by 3. This highlights a nonlinear relationship between price and quantity, meaning price changes at a changing rate as quantity demanded shifts.Understanding the demand equation is essential as it helps predict consumer behavior by showing how quantity demanded will adjust when there are alterations in price. For students studying economics, mastering this equation is crucial for calculating consumer surplus and other economic measures.

Integral calculus is a fundamental branch of calculus focused on accumulation and area under curves. In the context of economics, it's used to calculate total values like the consumer surplus by integrating the demand curve. The exercise provides a demand curve: \[ p = 100 - 3\sqrt{q} \]To find the consumer surplus, we integrate this demand equation with respect to quantity \(q\) from 0 to the quantity demanded at a specific price, \(\bar{q}\). The integration process captures the total area under the demand curve up to \(\bar{q}\). This total area represents the total willingness to pay by all consumers for \(\bar{q}\) units.Here is the integral calculation:\[ \int_{0}^{64} (100 - 3\sqrt{q}) \, dq \]Through substitution and integration techniques, you end up with:\[ \left(100q - 2q^{\frac{3}{2}}\right]_{0}^{64} = 5376 \]By understanding this aspect of calculus, students can calculate not just areas under curves but also real-world economic values.

Quantity demanded is a critical economic concept referring to the total amount of a good consumers are willing and able to purchase at a specific price during a given period. Determining the quantity demanded helps economists and businesses make informed decisions by predicting consumer purchasing behavior. In the provided exercise, to find the quantity demanded at the given price \(\bar{p} = 76\), you substitute into the demand equation:\[ 76 = 100 - 3\sqrt{\bar{q}} \]Solving the above equation lets you isolate and compute \(\bar{q}\) as follows:\[ \sqrt{\bar{q}} = 8 \]\[ \bar{q} = 8^2 = 64 \]Thus, at a price of 76, consumers are willing to buy 64 units of the product. Understanding this value is essential as it shows how much of a product the market will provide at a particular price point. This insight is vital for setting prices and forecasting demand in the marketplace.