Business Calculus - Math 1329 - Section 1.4 - Functional Models

This paragraph introduces the concept of creating functional models from word problem scenarios. The focus is on transforming a problem into a function to find various values such as the maximum or minimum of the function. An example of a farmer building a rectangular pen with a limited amount of fencing material is used to illustrate the process. The problem is broken down into geometric terms, and the relationship between the perimeter of the rectangle (600 feet) and the area it encloses is explored. The solution involves creating a function for the area in terms of one variable (X), which is then used to find the dimensions and area that maximize the pen's size.

The second paragraph continues the theme of optimizing rectangles but introduces a variation where the farmer builds a pen against a barn, requiring fencing for only three sides. The perimeter equation is adjusted accordingly (2x + y = 600), and the area function is manipulated to express y in terms of x. Using the vertex formula for a quadratic function, the maximum area is found, leading to the conclusion that a square provides the largest area. The dimensions and the largest possible area for the pen are calculated based on the given constraints.

The third paragraph shifts the focus to designing an open-top rectangular box with a fixed volume and specific width to minimize construction cost. The cost function includes the cost of material for the lateral sides and the base, with different costs per square foot. Using the volume constraint and the cost function, an equation relating the dimensions of the box to the cost is formulated. The minimum cost is found by graphing the function and using a graphing calculator to determine the dimensions that yield the minimum cost. The resulting box dimensions and the minimum cost are provided.

The fourth paragraph discusses the concept of market equilibrium using the example of two-liter soda production and sales. The supply and demand functions are given, and market equilibrium is defined as the point where supply equals demand. By setting the supply function equal to the demand function, the quantity that should be produced and sold for market equilibrium is found. The equilibrium price is then calculated by substituting the equilibrium quantity into either the supply or demand function. The resulting equilibrium price and quantity, along with a graphical representation of surplus and shortage scenarios, are presented.

The fifth paragraph delves into the graphical representation of market forces, specifically supply and demand curves, and how they interact to reach market equilibrium. The impact of different pricing levels on the market is explained, showing how prices can lead to surpluses or shortages. The process of finding the equilibrium point where the quantity demanded matches the quantity supplied is described. An example is given where the supply and demand functions for two-liter sodas are provided, and the calculation for market equilibrium in terms of quantity and price is demonstrated. The resulting equilibrium price and quantity are identified, and a graphical illustration of the supply and demand curves is included to show surplus and shortage areas.