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Chapter 2: Problem 8

One yard is equal to three feet. Measurement in y. a. Make a table like this showing the number of feet in lengths from 1 to 5 yards. b. For each additional yard in your table, how many more feet are there? c. Write a proportion that you could use to convert the measurements between \(y\) yards and \(f\) feet. d. Use the proportion you wrote to convert each measurement. 150 yards \(=f\) feet

Short Answer

Expert verified
150 yards is equal to 450 feet.

Step by step solution

01

Creating the Table

Let's start by creating a table of the number of feet corresponding to lengths from 1 to 5 yards. Given that 1 yard is equal to 3 feet, the table will list each respective yard and its equivalent in feet. | Yards (y) | Feet (f) | |-----------|----------| | 1 | 3 | | 2 | 6 | | 3 | 9 | | 4 | 12 | | 5 | 15 |
02

Analyzing the Increment

In the table, observe that as the number of yards increases by 1, the number of feet increases by 3. This is because each yard converts to 3 feet. So for each additional yard, there are 3 more feet.
03

Writing the Proportion

To write a proportion for converting yards to feet, we use the relationship: 1 yard equals 3 feet. This can be represented as a ratio:\[\frac{y}{1} = \frac{f}{3}\]This proportion can be used to find the number of feet, \(f\), given any number of yards, \(y\).
04

Using the Proportion for Conversion

Now, let's use the proportion to convert 150 yards into feet. Using the proportion \(\frac{y}{1} = \frac{f}{3}\), we replace \(y\) with 150:\[\frac{150}{1} = \frac{f}{3}\]Solving for \(f\), we multiply both sides by 3:\[f = 150 \times 3 = 450\]Therefore, 150 yards is equal to 450 feet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Measurement
Measurement is a fundamental concept in mathematics and science. It involves determining the size, length, or amount of something. In this exercise, we focus on measuring distances using yards and feet. One yard is defined as 3 feet. Understanding this conversion is essential when working with different units of measurement, especially in cases where precision is critical. For example, when working on projects or solving problems that involve distances, knowing how to change between units can help you complete tasks accurately. To measure properly, it's important to:
  • Understand the units being used.
  • Know the conversion rates, like 1 yard = 3 feet.
  • Create tables or use formulas for quick reference.
By mastering these basics, students can approach measurement-related problems with confidence.
Proportions
Proportions are a way to show the relationship between two quantities. They are particularly useful in unit conversion where one unit can be expressed as a multiple of another. In this exercise, a proportion helps us convert yards to feet using the known relationship that 1 yard equals 3 feet. The proportion can be set up as:\[\frac{y}{1} = \frac{f}{3}\]where \(y\) is the number of yards and \(f\) is the number of feet.To apply this proportion:
  • Insert the number of yards you want to convert for \(y\).
  • Solve for \(f\) by multiplying both sides by 3.
This proportion shows you exactly how many feet correspond to any number of yards, making conversions straightforward and efficient. By practicing with proportions, students gain a tool that simplifies many mathematical processes.
Linear Relationships
Linear relationships describe a consistent, proportional increase or decrease between two variables. In our problem, the relationship between yards and feet is linear. Each additional yard consistently adds three feet.This can be mathematically expressed with the equation:\[f = 3y\]This represents a direct relationship, where \(f\) (feet) increases linearly as \(y\) (yards) increases. In simpler terms:
  • If you know how many yards you have, you can predict the feet by multiplying yards by 3.
  • For every increase in yards, the feet increase by an equal amount (3 feet).
The simplicity of linear relationships makes them a powerful tool for solving problems involving proportional changes, as they provide a straightforward way to predict how variations in one quantity will affect another.

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