Problem 94 - A Graphical Approach to Precalculus with Limits

Question

Solve the following. Complete the following. (a) Write a function \(T\) that converts \(x\) tons to pounds. (b) Write a function \(O\) that converts \(x\) pounds to ounces. (c) Write a formula for the composition \((O \circ T)(x)\) (d) Explain what \((O \circ T)(x)\) calculates.

Step-by-Step Solution

Answer

(a) \( T(x) = 2000x \); (b) \( O(x) = 16x \); (c) \( O(T(x)) = 32000x \); (d) It converts tons to ounces.

Step 1: Understanding Tons to Pounds

The first task is to convert tons to pounds. We are given a conversion factor: 1 ton = 2000 pounds. Thus, our function \( T \) should reflect this conversion. The function for \( T \) will be \( T(x) = 2000x \), where \( x \) is the amount in tons.

Step 2: Understanding Pounds to Ounces

Next, we need to convert pounds to ounces. Knowing the conversion: 1 pound = 16 ounces, we can define our function \( O \). The function will be: \( O(x) = 16x \), where \( x \) refers to the weight in pounds.

Step 3: Composing the Functions

Now we'll find the composition of the two functions: \( (O \circ T)(x) \). This means we take the output of \( T(x) \) and use it as the input for \( O(x) \). The composed function is \( O(T(x)) = O(2000x) = 16(2000x) = 32000x \).

Step 4: Interpreting the Composition

The function \( (O \circ T)(x) = 32000x \) calculates the number of ounces in \( x \) tons. Essentially, it directly converts tons to ounces, knowing that 1 ton is equivalent to 32000 ounces (since 1 ton = 2000 pounds and 1 pound = 16 ounces).

Key Concepts

Unit Conversion

Unit conversion is an essential skill in math and daily life. It involves changing a quantity expressed in one unit to another, using a relationship between the two units.
For example, converting tons to pounds or pounds to ounces requires knowing the relationship between these units.
Unit conversion is vital in different fields such as engineering, cooking, and science.

• Allows for precision in measurement.
• Facilitates understanding by expressing quantities in familiar units.

Understanding how to move between units ensures accuracy and consistency in calculations. It's crucial to know the conversion factor, which is a constant value used to convert a measurement from one unit to another. Knowing how to apply these conversions is very useful for problem-solving.

Tons to Pounds

When converting tons to pounds, you apply a straightforward formula using a known conversion factor. Specifically, 1 ton is equal to 2000 pounds.
This conversion factor helps us create a function that makes the conversion seamless.

• Given: 1 ton = 2000 pounds.
• Conversion function: \( T(x) = 2000x \), where \( x \) is in tons.

Using this formula, you can easily convert any quantity from tons to pounds by simply multiplying the amount of tons by 2000.
For instance, if you have 3 tons, you would calculate the pounds as follows: \( T(3) = 2000 \, \times \, 3 = 6000 \) pounds.

Pounds to Ounces

To change pounds into ounces, we use another conversion factor. Here, 1 pound is equivalent to 16 ounces.
Converting between these units follows a similar straightforward process.

• Given: 1 pound = 16 ounces.
• Conversion function: \( O(x) = 16x \), where \( x \) is in pounds.

This means you multiply the number of pounds by 16 to get the number of ounces.
For example, if you have 8 pounds, the equivalent in ounces is \( O(8) = 16 \, \times \, 8 = 128 \) ounces.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario. By using functions like the ones we've developed, we can model the conversion of tons to ounces efficiently.
First, you convert tons to pounds using the function \( T(x) = 2000x \).
Next, you take that result and convert pounds to ounces using \( O(x) = 16x \).
Combining these steps, we get a composed formula \( (O \circ T)(x) = O(T(x)) = 32000x \).

• This composite function models the direct conversion from tons to ounces.
• It effectively calculates how many ounces are in a given number of tons.

Mathematical modeling through function composition simplifies complex calculations by breaking them down into smaller, manageable parts.
This approach makes solving real-world problems more intuitive and structured.