Problem 126 - Intermediate Algebra

Question

Simplify. See a Concept Check in this section. Assume variables represent positive numbers. The owner of Knightime Classic Movie Rentals has determined that the demand equation for renting older released DVDs is \(F(x)=0.6 \sqrt{49-x^{2}},\) where \(x\) is the price in dollars per two-day rental and \(F(x)\) is the number of times the \(\mathrm{DVD}\) is demanded per week. A. Approximate to one decimal place the demand per week of an older released DVD if the rental price is S3 per two-day rental. B. Approximate to one decimal place the demand per week of an older released DVD if the rental price is S5 per two-day rental. C. Explain how the owner of the store can use this equation to predict the number of copies of each DVD that should be in stock.

Step-by-Step Solution

Answer

A: 3.8 rentals; B: 2.9 rentals; C: Predict demand to manage stock.

Step 1: Substitute and Solve for A

Firstly, we need to find the demand when the rental price is \(3. Substitute \(x = 3\) into the demand equation \(F(x) = 0.6\sqrt{49 - x^2}\). This yields \(F(3) = 0.6\sqrt{49 - 3^2} = 0.6\sqrt{49 - 9} = 0.6\sqrt{40}\). Compute \(\sqrt{40}\), which approximately equals 6.32. Then calculate \(0.6 \times 6.32\), which equals approximately 3.8. Hence, the demand is 3.8 times per week for a \)3 rental.

Step 2: Substitute and Solve for B

Next, approximate the demand when the rental price is \(5. Substitute \(x = 5\) in the demand equation \(F(x) = 0.6\sqrt{49 - x^2}\). Thus, \(F(5) = 0.6\sqrt{49 - 5^2} = 0.6\sqrt{49 - 25} = 0.6\sqrt{24}\). Compute \(\sqrt{24}\), approximately equal to 4.9. Then, \(0.6 \times 4.9\) gives about 2.9. Therefore, the demand is approximately 2.9 times per week for a \)5 rental.

Step 3: Explain the Use of Equation

The owner can use this equation to predict the number of DVD rentals per week based on different price points. By substituting various values of \(x\), the owner can determine how price changes affect demand, helping them decide the optimal number of copies to keep in stock. This prediction helps maintain sufficient stock without over-purchasing.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra. Here, the given demand equation is simplified step by step for ease in calculations.The equation provided is:\[ F(x) = 0.6 \sqrt{49-x^2} \]To simplify expressions involving variables and constants, execute the following:

• Identify terms: Break down the equation into different terms. In our equation, it's \(49 - x^2\).
• Substitute known values: Replace variables with given numbers to simplify calculations.
• Perform arithmetic: Calculate the numerical value inside the expression’s root to continue simplification.

In the case of the demand equation, substituting \(x = 3\) transforms it into a straightforward computation:\[ F(3) = 0.6 \sqrt{49 - 9} = 0.6 \sqrt{40} \]Approximating the terms within simplifies the expression, paving the way for further calculations, as we've seen in the problem solution.

Square Root Calculations

Calculating square roots accurately is vital in many mathematical scenarios, including demand equations, like the one provided in our problem.The equation uses a square root to determine the demand:\[ F(x) = 0.6 \sqrt{49-x^2} \]When calculating square roots:

• Estimate and approximate: Not all square roots are perfect squares; approximation helps in these cases.
• Use a calculator: For awkward numbers, rely on calculators to get precise square root values.

For instance, in the demand equation, calculate \(\sqrt{40}\) after substitution, which approximates to \(6.32\). Then, multiply by \(0.6\), yielding a refined demand estimate. Having a solid grasp of square root calculations aids in accurate demand predictions and general problem-solving.

Predicting Demand

Predicting demand involves using equations and data to estimate consumer interest.Consider the demand equation from the exercise:\[ F(x) = 0.6 \sqrt{49-x^2} \]This equation aids in predicting how many times a DVD will be rented weekly based on price \(x\):

• Plug and predict: Substitute price values into the equation to see their effect on demand.
• Adjust strategies: Use these predictions to inform stocking decisions and pricing strategies.

For example, plugging \(x = 3\) resulted in a demand of about \(3.8\), whereas \(x = 5\) gave approximately \(2.9\). Using these predictions helps the business plan inventory based on pricing adjustments to meet customer demand efficiently.

Price Elasticity

Price elasticity measures how sensitive the demand for a product is to price changes.In the case of the DVD rental scenario, the demand equation makes it evident:\[ F(x) = 0.6 \sqrt{49-x^2} \]Understanding price elasticity through this equation helps a business owner to:

• Determine sensitivity: Observe changes in demand as \(x\), the price, varies.
• Maximize revenue: Adjust prices to find a balance between price and demand to optimize sales.

By substituting different prices into the equation and noting the corresponding changes in demand, the owner can estimate the price elasticity of their rentals.
For instance, the decrease in demand from \(3.8\) to \(2.9\) as the price increases from \(3\) to \(5\) illustrates demand sensitivity.This concept helps refine pricing strategies for achieving desired financial outcomes while maintaining appropriate stock levels.