Problem 218 - Calculus Volume 1

Question

A function that converts dress sizes in the United States to those in Europe is given by \(D(x)=2 x+24\) . a. Find the European dress sizes that correspond to sizes \(6,8,10,\) and 12 in the United States. b. Find the function that converts European dress sizes to U.S. dress sizes. c. Use part b. to find the dress sizes in the United States that correspond to \(46,52,62,\) and \(70 .\)

Step-by-Step Solution

Answer

US sizes 6, 8, 10, 12 convert to EU sizes 36, 40, 44, 48. The inverse function is \(x = \frac{E - 24}{2}\). EU sizes 46, 52, 62, 70 convert to US sizes 11, 14, 19, 23.

Step 1: Finding European Size for US Size 6

Consider the function provided: \(D(x) = 2x + 24\). Insert the US size 6 into the function. \[D(6) = 2(6) + 24\] Evaluate the calculation: \[D(6) = 12 + 24 = 36\] Thus, a US size 6 is equivalent to a European size 36.

Step 2: Finding European Size for US Size 8

Using the same function \(D(x) = 2x + 24\), substitute US size 8 in: \[D(8) = 2(8) + 24\] Calculate the value: \[D(8) = 16 + 24 = 40\] Therefore, a US size 8 corresponds to a European size 40.

Step 3: Finding European Size for US Size 10

Insert US size 10 into the function \(D(x) = 2x + 24\): \[D(10) = 2(10) + 24\] Perform the calculation: \[D(10) = 20 + 24 = 44\] US size 10 is equivalent to European size 44.

Step 4: Finding European Size for US Size 12

Substitute US size 12 into the function, \(D(x) = 2x + 24\): \[D(12) = 2(12) + 24\] Compute the result: \[D(12) = 24 + 24 = 48\] Hence, a US size 12 translates to a European size 48.

Step 5: Finding the Inverse Function

We want the function that converts European sizes back to US sizes. Start with the given function: \(E = 2x + 24\). To find \(x\), solve for \(x\): \[E = 2x + 24\] Rearrange to find \(x\): \[2x = E - 24\] Divide by 2 to isolate \(x\): \[x = \frac{E - 24}{2}\] Therefore, the inverse function is \(x = \frac{E - 24}{2}\).

Step 6: Finding US Size for European Size 46

Using the inverse function \(x = \frac{E - 24}{2}\), replace \(E\) with 46: \[x = \frac{46 - 24}{2}\] Calculate the result: \[x = \frac{22}{2} = 11\] European size 46 corresponds to US size 11.

Step 7: Finding US Size for European Size 52

Use the inverse function: \(x = \frac{E - 24}{2}\). Insert \(E = 52\): \[x = \frac{52 - 24}{2}\] Evaluate the expression: \[x = \frac{28}{2} = 14\] Therefore, a European size 52 converts to US size 14.

Step 8: Finding US Size for European Size 62

With the inverse function \(x = \frac{E - 24}{2}\), substitute \(E = 62\): \[x = \frac{62 - 24}{2}\] Compute: \[x = \frac{38}{2} = 19\] European size 62 corresponds to US size 19.

Step 9: Finding US Size for European Size 70

Apply the inverse function: \(x = \frac{E - 24}{2}\), using \(E = 70\): \[x = \frac{70 - 24}{2}\] Perform the calculation: \[x = \frac{46}{2} = 23\] Thus, European size 70 is equal to US size 23.

Key Concepts

Inverse Functions

The idea of inverse functions plays a crucial role in calculus and transformations. An inverse function essentially reverses the work done by a function. If we have a function that maps input values to outputs, like converting U.S. dress sizes to European sizes with the formula \( D(x) = 2x + 24 \), the inverse function lets us perform the conversion in the opposite direction.
This means for every European size, we can find the original U.S. size. To derive the inverse, we set the equation equal to the European size \( E \) and solve for the U.S. size \( x \):
\[E = 2x + 24\]Rearranging gives us:

• Subtract 24 from both sides: \( 2x = E - 24 \)
• Divide by 2: \( x = \frac{E - 24}{2} \)

Thus, the inverse function \( x = \frac{E - 24}{2} \) enables us to determine U.S. sizes from European sizes efficiently.

Function Evaluation

Function evaluation is a fundamental concept in mathematics where specific values are inserted into a function to obtain a result. The process involves substituting the input, often represented as variable \( x \), with given values and following the operations dictated by the function's expression.
This exercise's goal is to take U.S. dress sizes and find their equivalent European sizes using the conversion function \( D(x) = 2x + 24 \). For instance, substituting \( x = 6 \), we calculate as follows:

• \( D(6) = 2(6) + 24 = 36 \)

This tells us that a U.S. size 6 corresponds to a European size 36. By continuing this pattern for other values, we can establish an understanding and develop efficiency in evaluating functions.

Mathematical Modeling

Mathematical modeling enables us to create a representation of real-world situations using mathematical concepts. In this problem, the relationship between U.S. dress sizes and European dress sizes has been depicted by a linear function:
\[ D(x) = 2x + 24 \]
This equation models how changes in U.S. sizes correspond linearly to shifts in European sizes. By adjusting the input, we recognize a consistent calculation pattern where each increase by one U.S. size results in an increase of two in the European size plus a constant offset of 24.
This type of modeling is advantageous because it gives a clear formula to apply to any U.S. size, making the conversion quick and accurate.

Conversion Functions

Conversion functions are designed to translate one set of measurements or units into another. They are practical tools often used in multiple contexts, such as clothing sizes, temperature scales, or currency exchange.
In this exercise, we see a conversion function that moves between U.S. and European dress sizes: \( D(x) = 2x + 24 \). This equation implies that irrespective of the U.S. size, there is a defined mathematical operation by which one can determine its European counterpart.

• This conversion follows a linear pattern, indicating predictability.
• The reverse conversion using the inverse function \( x = \frac{E - 24}{2} \) highlights how one formula can yield two-way conversions.

Understanding and applying conversion functions solidifies our comprehension of transformation between varying units and systems, ensuring solutions are systematic and reliable.