Problem 57 - Algebra A Combined Function

Question

The function \(f(x)=0.19 x^{2}+5.67 x+43.7\) can be used to approximate the growth of restaurant food-and-drink sales, where \(x\) is the number of years since 1970 and \(f(x)\) or \(y\) is the sales (in billions of dollars.) a. Approximate the restaurant food-and-drink sales in 2005 . b. Approximate the restaurant food-and-drink sales in 2010 . c. Use this function to estimate the restaurant food-and-drink sales in \(2015 .\) d. From parts (a), (b), and (c), determine whether the restaurant food-and- drink sales are increasing at a steady rate. Explain why or why not.

Step-by-Step Solution

Answer

Sales increase but not at a steady rate due to quadratic growth.

Step 1: Determine Years Since 1970

To use the given function, we need to convert the years provided into the number of years since 1970. For 2005, the number of years since 1970 is obtained by subtracting 1970 from 2005. Similarly, for 2010 and 2015, subtract 1970 from those years to find the values of \(x\) to use in the function.

Step 2: Plug in Values to the Function

Use the function \(f(x) = 0.19x^2 + 5.67x + 43.7\) to find the sales for the years identified in Step 1. Calculate \(f(35)\), \(f(40)\), and \(f(45)\) for the years 2005, 2010, and 2015 respectively.

Step 3: Calculate Sales for 2005

Plug \(x = 35\) into the function to find the sales for 2005: \[f(35) = 0.19(35)^2 + 5.67(35) + 43.7\]Calculate \(f(35)\) to get the approximate sales for 2005.

Step 4: Calculate Sales for 2010

Plug \(x = 40\) into the function to find the sales for 2010: \[f(40) = 0.19(40)^2 + 5.67(40) + 43.7\]Calculate \(f(40)\) to get the approximate sales for 2010.

Step 5: Calculate Sales for 2015

Plug \(x = 45\) into the function to find the sales for 2015: \[f(45) = 0.19(45)^2 + 5.67(45) + 43.7\]Calculate \(f(45)\) to get the approximate sales for 2015.

Step 6: Analyze Sales Growth Rate

Compare the results from Steps 3, 4, and 5 to determine if sales are increasing at a steady rate. Calculate the differences between yearly sales and discuss the pattern of growth based on the results.

Key Concepts

Function Approximation

When talking about function approximation, it refers to using a mathematical function to estimate or predict other values. In our case, the function \(f(x)=0.19 x^{2}+5.67 x+43.7\) helps us approximate restaurant sales. Function approximation is crucial because it allows us to use a simple model to make predictions about real-world phenomena. This usually involves fitting a function to historical data to attempt to capture and predict the trend.
Additionally, this function we have captures the sales growth over the years since 1970. Thus, it uses an easily inputted variable, \(x\), which signifies the years since 1970. By inputting the specific number of years elapsed, we can approximate what the sales would look like in a particular future year.

Sales Forecasting

Sales forecasting involves predicting future sales using historical data and trends. Here, our quadratic function \(f(x)\) helps estimate sales figures for specific years such as 2005, 2010, and 2015. Forecasting is vital for businesses because it helps in decision-making regarding resource allocation, budgeting, and strategy development.

• For 2005, we find \(f(35)\), which gives our sales estimate for that year.
• Then, for 2010, \(f(40)\) helps us predict sales for that year.
• Finally, \(f(45)\) allows us to estimate sales for 2015.

Since the function is designed to approximate sales, it is naturally embedded with the trend the historical data exhibited, although one should always understand limitations and possible deviations in real scenarios.

Growth Analysis

In growth analysis, we determine if and how sales figures are increasing. By plugging different values of \(x\) into our function and observing the results, we analyze whether sales are growing steadily.
After calculating \(f(35)\), \(f(40)\), and \(f(45)\), we compare these sales figures. The differences between these values showcase sales growth over time. It’s essential to evaluate if the increments are uniform or increasing at an accelerating rate.

• A steady growth would imply consistent increment each year.
• An accelerating growth would mean the annual changes grow larger each year.

This analysis helps in understanding overall trends and whether investments are yielding the desired outcomes.

Algebraic Modeling

Algebraic modeling involves crafting algebraic expressions and equations to represent real-world situations. In our exercise, we model the restaurant sales data using a quadratic function.

• This form of modeling provides a structured way to represent data trends.
• It allows for efficient computation, as we can easily plug in numbers to predict sales in the future.
• Quadratic functions, like this one, represent non-linear growth, implying compounding effects which are common in economic contexts.

Algebraic modeling is powerful as it helps translate complex real-world behaviors into manageable and analyzable mathematical expressions. This essential tool aids in scenario planning and helps stakeholders make data-driven decisions.