Problem 62 - Precalculus with Limits

Question

Supply and Demand The supply and demand curves for a business dealing with wheat are Supply: \(p=1.45+0.00014 x^{2}\) Demand: \(p=(2.388-0.007 x)^{2}\) where \(p\) is the price in dollars per bushel and \(x\) is the quantity in bushels per day. Use a graphing utility to graph the supply and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for \(x>0 . )\)

Step-by-Step Solution

Answer

The market equilibrium is the point where the graphs of the supply and demand curves intersect. This point would best be calculated using a graphing utility and identifying the intersection point from the graph.

Step 1: Understanding the Equations

Firstly, the given formulas represent the supply and demand curves. The supply curve is given by \(p=1.45+0.00014 x^{2}\) and the demand curve by \(p=(2.388-0.007 x)^{2}\). In both formulas, 'p' represents the price per bushel and 'x' is the quantity of bushels sold per day.

Step 2: Setting the Equations Equal to Each Other

To find the market equilibrium, set the supply and demand equations equal to each other. So, the equation becomes \(1.45+0.00014 x^{2} = (2.388-0.007 x)^{2}\).

Step 3: Using a Graphing Utility

Next, input both functions into a graphing utility to visualize their behavior. The point where these two curves intersect is the market equilibrium.

Step 4: Solving for Intersection

Find the intersection of the two graphs. This may require approximation or use of numerical methods if solving analytically is not possible.

Step 5: Interpreting Results

The 'x' coordinate of the intersection point represents the quantity of bushels sold per day at equilibrium, and the 'p' coordinate represents the price per bushel at which goods are sold/bought in equilibrium.

Key Concepts

Supply and Demand Curves

The supply and demand curves are fundamental concepts in economics, representing how price varies with quantity supplied and demanded. In this exercise, the supply curve is described by the equation \( p = 1.45 + 0.00014 x^2 \). This equation indicates that as the quantity of wheat increases, the price also slightly increases, which is typical since suppliers may need higher prices to provide more goods. On the other hand, the demand curve \( p = (2.388 - 0.007 x)^2 \) shows an inverse relationship, where as the quantity increases, the price consumers are willing to pay decreases, reflecting the law of demand. This exercise challenges us to understand these relationships and how they interact to determine the market equilibrium price and quantity.

Graphing Utility

Using a graphing utility can be incredibly helpful in visualizing supply and demand curves. Graphing utilities are tools that allow for the visualization of complex equations and their intersections, facilitating a deeper understanding and easier analysis. By plotting the supply curve \( p = 1.45 + 0.00014 x^2 \) and the demand curve \( p = (2.388 - 0.007 x)^2 \) on a graph, students can visually identify the point of intersection. This graphical solution provides an intuitive way to understand the market equilibrium. Tools like graphing calculators or online graphing software can automatically generate accurate graphs based on given equations, helping students see the balance point where supply equals demand.

Intersection of Graphs

The intersection of the supply and demand graphs represents the market equilibrium. This pivotal point is where the quantity of wheat that suppliers want to sell equals the quantity consumers want to purchase at the equilibrium price. To find this point, both equations \( 1.45 + 0.00014 x^2 = (2.388 - 0.007 x)^2 \) are set equal. The graphical representation highlights exactly where these curves meet. This intersection is crucial because it indicates the optimal price and quantity where the market is most efficient and neither excess surplus nor shortage occurs. Understanding this intersection helps in grasping how real-world markets operate and adjust to achieve balance.

Numerical Methods

In cases where analytical solutions are complex or impractical, numerical methods provide an alternative way to find the intersection of curves. This exercise may require approximation techniques to solve \( 1.45 + 0.00014 x^2 = (2.388 - 0.007 x)^2 \) accurately. Numerical methods, such as the Newton-Raphson method or using iterative algorithms, can be employed to estimate the solution. These techniques involve making educated guesses for the values and repeatedly refining these guesses to arrive at precise results. While graphing tools give a visual approximation, numerical methods allow for a more exact calculation. Learning these methods is essential for tackling real-world problems where straightforward solutions aren't always possible.