Problem 43 - Calculus for Biology and Medicine

Question

To convert a length measured in feet to a length measured in centimeters, we use the facts that a length measured in feet is proportional to a length measured in centimeters and that \(1 \mathrm{ft}\) corresponds to \(30.5 \mathrm{~cm}\). If \(x\) denotes the length measured in \(\mathrm{ft}\) and \(y\) denotes the length measured in \(\mathrm{cm}\), then $$ y=30.5 x $$ (a) Explain how to use this relationship. (b) Use the relationship to convert the following measurements into centimeters: (i) \(6 \mathrm{ft}\) (ii) \(3 \mathrm{ft}, 2 \mathrm{in}\) (iii) \(1 \mathrm{ft}, 7\) in (c) Use the relationship to convert the following measurements into ft: (i) \(173 \mathrm{~cm}\) (ii) \(75 \mathrm{~cm}\) (iii) \(48 \mathrm{~cm}\)

Step-by-Step Solution

Answer

(a) Use y = 30.5x to convert feet to cm by multiplying x by 30.5. (b) (i) 183 cm, (ii) 96.75 cm, (iii) 48.29 cm. (c) (i) 5.672 ft, (ii) 2.459 ft, (iii) 1.574 ft.

Step 1: Understanding the Proportionality Relationship

The equation given is \( y = 30.5x \), where \( x \) is the length in feet and \( y \) is the length in centimeters. This means that to convert any measurement from feet to centimeters, you simply multiply the number of feet by 30.5, since there are 30.5 centimeters in one foot. This equation also implies that if you know the length in centimeters, you can find the length in feet by dividing by 30.5.

Step 2: Convert 6 Feet to Centimeters

To convert \(6 \text{ ft} \) to centimeters, substitute \( x = 6 \) in the equation \( y = 30.5x \). Calculate \( y = 30.5 imes 6 = 183 \). Thus, \( 6 \text{ ft} \) is equivalent to \( 183 \text{ cm} \).

Step 3: Convert 3 Feet, 2 Inches to Centimeters

First, convert the inches to feet: 2 inches is \( \frac{2}{12} = \frac{1}{6} \) feet. Add this to 3 feet to get a total of \( 3 + \frac{1}{6} = \frac{19}{6} \) feet. Now convert this total to centimeters: \( y = 30.5 \times \frac{19}{6} = 96.75 \text{ cm} \). Hence, \( 3 \text{ ft, } 2 \text{ in} \) is equivalent to \( 96.75 \text{ cm} \).

Step 4: Convert 1 Foot, 7 Inches to Centimeters

First, convert the inches to feet: 7 inches is \( \frac{7}{12} \) feet. Add this to 1 foot to get a total of \( 1 + \frac{7}{12} = \frac{19}{12} \) feet. Now convert this total to centimeters: \( y = 30.5 \times \frac{19}{12} = 48.29 \text{ cm} \). Thus, \( 1 \text{ ft, } 7 \text{ in} \) is equivalent to \( 48.29 \text{ cm} \).

Step 5: Convert 173 cm to Feet

Using the inverse relationship, convert centimeters to feet. For \( 173 \text{ cm} \), calculate \( x = \frac{173}{30.5} \approx 5.672 \). Therefore, \( 173 \text{ cm} \) is approximately \( 5.672 \text{ ft} \).

Step 6: Convert 75 cm to Feet

For \( 75 \text{ cm} \), divide by 30.5: \( x = \frac{75}{30.5} \approx 2.459 \). Hence, \( 75 \text{ cm} \) is approximately \( 2.459 \text{ ft} \).

Step 7: Convert 48 cm to Feet

For \( 48 \text{ cm} \), divide by 30.5: \( x = \frac{48}{30.5} \approx 1.574 \). Thus, \( 48 \text{ cm} \) is approximately \( 1.574 \text{ ft} \).

Key Concepts

Proportionality

Proportionality is a fundamental concept that describes a consistent relationship between two quantities. In the context of unit conversion, if you alter one quantity, the second changes in a predictable way based on their proportional relationship.

For example, when converting lengths between feet and centimeters, each foot corresponds to 30.5 centimeters. This is a classic example of proportionality. If you know the length in feet (let's call it \( x \)), you can calculate the length in centimeters (\( y \)) as \( y = 30.5 \times x \).
This equation shows that the centimeter measurement is directly proportional to the feet measurement by a factor of 30.5. This means that if you double the number of feet, you also double the number of centimeters, maintaining the same proportion.

Length Measurement

Length measurement is the process of determining the size or extent of something from end to end. It is a way to quantify the physical dimension of an object, allowing for accurate descriptions and comparisons.

There are different units for measuring length, and they can be interconverted using known conversion ratios, like from feet to centimeters.

For example, imagine you are given a measurement in feet and need to express it in centimeters. Knowing that 1 foot is equivalent to 30.5 centimeters facilitates this conversion. You simply multiply the number of feet by 30.5 to find the length in centimeters.

• 6 feet becomes \(6 \times 30.5 = 183 \) centimeters.
• 3 feet, 2 inches is converted by first changing the inches to feet, then multiplying.
• Measurement in centimeters can be converted back to feet by dividing by 30.5.

The ability to measure and convert lengths reliably is crucial in various fields, such as construction, engineering, and many everyday applications.

Mathematical Equations

Mathematical equations are statements that express the equality of two expressions. They are used to represent relationships between variables and to perform conversions between units, among other things. In this exercise, we see the equation \( y = 30.5x \), which is used for unit conversion from feet to centimeters.

This equation plays a pivotal role in defining the conversion process:

• To convert feet into centimeters, identify \( x \) as the number of feet, then use the equation to calculate \( y \), which is the corresponding length in centimeters.
• Conversely, to convert centimeters to feet, rearrange the equation to solve for \( x \), giving \( x = \frac{y}{30.5} \). This shows inverse proportionality.

Solving these equations ensures accurate unit conversions and demonstrates how mathematical equations function as critical tools for clarifying and computing quantitative relationships.