Problem 11 - College Algebra with Modeling and Visualization

Question

Exercises 11 and 12: Write a symbolic representation (formula) for a function \(f\) that computes the following. (a) The number of pounds in \(x\) ounces (b) The number of dimes in \(x\) dollars (c) The monthly electric bill in dollars if \(x\) kilowattbours are used at 6 cents per kilowatt-hour and there is a fee of \(\$ 6.50\) (d) The cost of skiing \(x\) times with a \(\$ 500\) season pass

Step-by-Step Solution

Answer

(a) \(f_1(x) = \frac{x}{16}\), (b) \(f_2(x) = 10x\), (c) \(f_3(x) = 0.06x + 6.50\), (d) \(f_4(x) = 500\).

Step 1: Understanding the Problem

We need to develop symbolic formulas for given real-world relationships. We'll define each function based on the problem description.

Step 1: Pounds in Ounces

One pound is equivalent to 16 ounces. So, to convert ounces to pounds, divide by 16. Thus, the function is given by:\[ f_1(x) = \frac{x}{16} \] where \(x\) is the number of ounces.

Step 2: Dimes in Dollars

One dollar is equivalent to 10 dimes. To find the number of dimes in dollars, multiply by 10. Thus, the function is:\[ f_2(x) = 10x \] where \(x\) is the number of dollars.

Step 3: Monthly Electric Bill

The price of electricity per kilowatt-hour is 6 cents, which is \(0.06. The total cost is the product of the kilowatt-hours used and this price, plus a base fee of \)6.50. Thus, the function is:\[ f_3(x) = 0.06x + 6.50 \] where \(x\) is the number of kilowatt-hours.

Step 4: Skiing with a Season Pass

The cost is constant if there's a season pass, irrespective of the number of times you ski. Hence, the formula is simply the cost of the pass:\[ f_4(x) = 500 \] where \(x\) is the number of skiing events.

Key Concepts

Unit Conversion

Unit conversion involves changing a quantity expressed in one unit to another unit, often using a known conversion factor. This is crucial in math and science for simplifying and understanding measurements. In the exercise, for example, we needed to convert ounces to pounds.

To perform this conversion, you need to know that:

• 1 pound = 16 ounces.

To find how many pounds are in "x" ounces, you divide the number of ounces by 16. Thus, our conversion formula becomes:\[f_1(x) = \frac{x}{16}\] Here, we see a simple division operation at play. This helps us use a common and understandable measurement, like pounds, when we start from ounces—a fantastic practice in understanding and applying real-world relationships.

Cost Calculation

Cost calculation is a fundamental part of daily life which involves computing total expenses based on different factors. In the exercise, two examples demonstrated this concept clearly.

The first example discusses how one would calculate the electric bill using a given cost per kilowatt-hour and a standard fee. The calculation is straightforward:

• Cost per kilowatt-hour (6 cents) converted to dollars: 0.06.
• A base fee associated with the electric service: \$6.50.

So, the formula becomes:\[f_3(x) = 0.06x + 6.50\] This formula calculates a precise monthly bill based on actual kilowatt-hour usage, factoring in both the variable cost (usage) and fixed cost (base fee).

The second example shows how cost is simplified with a season pass where the cost remains constant regardless of usage. Here the cost of skiing is a one-time fee represented as:\[f_4(x) = 500\] This cost calculation reflects another real-world scenario where expenditure is not dependent on frequency or usage.

Real-World Relationships

Real-world relationships in algebra refer to expressing everyday scenarios using mathematical functions. These mathematical models allow us to predict and understand various outcomes based on changing conditions.

In the exercise, several everyday situations were expressed as functions. For instance:

• Ounces to pounds; needing a conversion factor due to the relationship between these units.
• Dollars to dimes; showing how currency can be broken down into smaller units of value.

These formulas show how algebra is used to represent and solve real-world problems. The core idea is to take something complex—weighing objects or breaking money into smaller change—and present it as a simple relationship or formula.

Symbolic Representation

Symbolic representation involves using functions or formulas to express mathematical ideas concisely. It's a powerful tool in algebra for summarizing complex ideas in simple mathematical terms.

In the exercise, symbolic representations were developed for various scenarios:

• Pounds in ounces: \(f_1(x) = \frac{x}{16}\)
• Dimes in dollars: \(f_2(x) = 10x\)
• Monthly electric bill: \(f_3(x) = 0.06x + 6.50\)
• Cost of skiing: \(f_4(x) = 500\)

Each function \(f_i(x)\) describes a real-world situation in a straightforward manner, translating practical problems into mathematical expressions. This makes them more manageable to interpret and solve effectively, showcasing algebra's capacity to handle real-world complexities seamlessly.