Problem 77 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The quantity \(x\) of satellite radios that a manufacturer will make available in the marketplace is related to the unit price \(p\) (in dollars) by the equation $$ p=\frac{1}{10} \sqrt{x}+10 $$ How many satellite radios will the manufacturer make available in the marketplace if the unit price is \(\$ 30\) ?

Step-by-Step Solution

Answer

The manufacturer will make 40,000 satellite radios available in the marketplace when the unit price is $30.

Step 1: Write down the given equation

The given equation relating the quantity of satellite radios \(x\) and the unit price \(p\) is: \( p=\frac{1}{10} \sqrt{x}+10 \)

Step 2: Substitute the given unit price

We are given the unit price \(p = 30\). We need to substitute this value into the given equation and solve for the quantity \(x\): \( 30 = \frac{1}{10} \sqrt{x} + 10 \)

Step 3: Solve for x

Now we need to isolate the term with \(x\) and solve for \(x\). \( \begin{aligned} 30 - 10 &= \frac{1}{10}\sqrt{x} \\ 20 &= \frac{1}{10}\sqrt{x} \\ 200 &= \sqrt{x} \\ x &= 200^2 \\ x &= 40000 \end{aligned} \)

Step 4: Conclusion

The manufacturer will make 40,000 satellite radios available in the marketplace when the unit price is $30.

Key Concepts

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent real-world situations. In the case of the satellite radio problem, a mathematical model is used to describe the relationship between the quantity of radios and their unit price. This allows manufacturers, decision-makers, and analysts to make predictions and decisions based on numerical evidence. By translating a real-world scenario into an equation, stakeholders can analyze various aspects of a production process without solely relying on trial and error. With the given equation \( p=\frac{1}{10} \sqrt{x}+10 \), we see how price (\(p\)) changes with the quantity (\(x\)) made available.\\Using mathematical modeling, businesses can determine how to maximize profit, minimize costs, or optimize production levels by tweaking variables like price and production quantity. Thus, mathematical modeling is a powerful tool in decision-making and strategic planning.

Problem Solving

Problem solving involves critical thinking and a structured approach to find a solution to a given problem. The satellite radio problem requires solving for the quantity of radios given a specific unit price. Here, a systematic approach is used in steps to tackle the problem effectively.\\

• First, clearly understand the problem by identifying the given and required variables. In this case, we know the unit price \(p = 30\) and need to find \(x\).
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• Next, apply the appropriate mathematical methods, such as substituting the known values into the given equation.
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• The final step is computing the solution and verifying its correctness, allowing us to conclude that the manufacturer will produce 40,000 radios.

\\By breaking down the problem into manageable steps, we can solve even complex problems efficiently. Practice in problem solving enhances analytical skills, making it easier to apply similar strategies to different scenarios.

Unit Price Equation

The unit price equation in this problem is \( p=\frac{1}{10} \sqrt{x}+10 \). This equation demonstrates how unit price is dependent on the quantity of goods made available in the market. Let's break down what this means and how it's applied.\\

• The equation establishes a functional relationship between price and quantity. The square root function indicates that price increases at a decreasing rate as more units are produced and made available. This reflection of real-world price dynamics shows diminishing returns with increased production.
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• In practice, this relationship helps businesses understand pricing strategies. If a specific price point is desired, like \$30 in our exercise, the equation can be manipulated (as done in the step-by-step solution) to find the corresponding quantity of goods to produce.
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• Understanding and utilizing such equations allow businesses to fine-tune their supply to meet market demand, seek competitive advantages, and adjust prices dynamically to market changes.

\\Unit price equations like these are fundamental for any business engaged in the production and sale of goods, highlighting the intersection of mathematics, economics, and business strategy.