Problem 84 - Precalculus with Limits

Question

The demand equation for a smart phone is \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\). Find the demand \(x\) for a price of \((\) a \() p=\$ 169\) and (b) \(p=\$ 299 .\)

Step-by-Step Solution

Answer

For a price of \$169, the demand is \[x=-\frac{1}{0.002}\ln\left(\frac{4}{1-\frac{169}{5000}}-4\right)\]. For a price of \$299, the demand is \[x=-\frac{1}{0.002}\ln\left(\frac{4}{1-\frac{299}{5000}}-4\right)\]. The exact numerical results will depend on the logarithm values.

Step 1: Understanding the Demand Equation

The demand equation for a specific product is given as \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\). In this equation, \(p\) represents the price of the product and \(x\) represents the demand for the product.

Step 2: Solving for (a) p=$169

Substitute \(p=169\) into the equation and solve for \(x\). This involves isolating \(x\) on one side of the equation. Start by simplifying the equation to give\[\frac{4}{1-\frac{p}{5000}}-4=e^{-0.002 x}\]then taking the natural logarithm on both sides and rearranging to give\[x=-\frac{1}{0.002}\ln\left(\frac{4}{1-\frac{p}{5000}}-4\right)\].Substituting in the given value for \(p\), we find that\[x=-\frac{1}{0.002}\ln\left(\frac{4}{1-\frac{169}{5000}}-4\right).\]

Step 3: Solving for (b) p=$299

Using the same approach as in step 2, substitute \(p=299\) into the equation. This gives\[x=-\frac{1}{0.002}\ln\left(\frac{4}{1-\frac{299}{5000}}-4\right).\]

Key Concepts

Economics in Mathematics

Mathematics is instrumental in economics for modeling relationships between different variables. For instance, the demand equation in the exercise is a critical mathematical concept in economics. It represents the relationship between the price of a product (\(p\)) and the quantity of that product demanded by consumers (\(x\)). Demand equations help economists and businesses understand consumer behavior, set pricing strategies, and predict market trends.

In this specific problem, the demand equation for a smartphone is an example of how mathematical models are used to capture complex economic realities, such as how demand can change with price adjustments. Understanding not only the equation itself but also how to interpret and solve it is essential for making informed economic decisions. By solving for different values of price, businesses can estimate the corresponding demand, which enables them to establish appropriate production levels and pricing policies to maximize revenue.

Logarithmic Functions

Logarithmic functions play a pivotal role in solving equations where the variable is in an exponent, which is the case in many real-world applications including economics. The function \(\ln(x)\) is the natural logarithm and it is the inverse of the exponential function \(e^x\). This means that logarithms can be used to 'undo' the effect of an exponentiation, and this characteristic is what makes them so powerful for solving exponential equations.

In the given exercise, after rearranging the equation to isolate the exponential term, we apply the natural logarithm on both sides to extract the variable from the exponent. This mathematical operation allows us to solve for \(x\), which represents the quantity demanded at a specific price point. Understanding how to work with logarithmic functions, including their laws and properties, is fundamental for students tackling real-world exponential problems in fields such as economics.

Solving Exponential Equations

An exponential equation involves a variable in the exponent, which can sometimes make them tricky to solve. The approach often involves employing logarithms to linearize the exponential nature of the equation, as logarithms provide the means to bring the exponent down to where we can manipulate and solve for it.

In the exercise, we are presented with an example where the demand (\(x\)) exponentially affects the price (\(p\)) through a base of \(e\), Euler's number, a fundamental constant in mathematics. Solving such equations correctly is crucial to find the demand for different prices. The specific method used in this solution, isolating the exponential term followed by the application of the natural logarithm, showcases a general approach to handling such equations. By mastering the method of solving exponential equations, students not only grasp the mathematical concept but also learn to tackle similar problems found in economic scenarios and beyond.