Problem 49 - Algebra 2

Question

MARATHONS The length of a marathon was determined in the 1908 Olympic Games in London, England. The race began at Windsor Castle and ended in front of the royal box at London's Olympic Stadium, which was a distance of 26 miles 385 yards. Determine how many feet the marathon covers using the formula \(f(m, y)=5280 m+3 y,\) where \(m\) is the number of miles and \(y\) is the number of yards.

Step-by-Step Solution

Answer

The marathon covers 138,435 feet.

Step 1: Identify Given Values

The problem states that the marathon covers 26 miles and 385 yards. We can identify these given values as \( m = 26 \) and \( y = 385 \).

Step 2: Understand the Formula

The formula given to convert miles and yards into feet is \( f(m, y) = 5280m + 3y \), where \( m \) is the number of miles and \( y \) is the number of yards.

Step 3: Substitute Values into the Formula

Substitute the values \( m = 26 \) and \( y = 385 \) into the formula: \[ f(26, 385) = 5280 \times 26 + 3 \times 385 \]

Step 4: Calculate Miles to Feet Conversion

First, calculate \( 5280 \times 26 \): \[ 5280 \times 26 = 137280 \] This converts the miles to feet.

Step 5: Calculate Yards to Feet Conversion

Next, calculate \( 3 \times 385 \): \[ 3 \times 385 = 1155 \] This converts the yards to feet.

Step 6: Sum the Feet from Miles and Yards

Add the two results from the previous steps to find the total feet: \[ 137280 + 1155 = 138435 \]

Key Concepts

Miles to Feet Conversion

Converting measurements is a vital skill in mathematics and everyday life. When looking into the conversion of miles to feet, it's important to remember the key conversion factor: there are 5,280 feet in one mile.
Understanding this conversion allows us to compute distances commonly expressed in miles and change them to feet, which can sometimes be more intuitive or applicable.

• Start with the given number of miles, for example, 26 miles as in our marathon problem.
• To convert this to feet, multiply the number of miles by the conversion factor (5,280):
  \[26 \times 5,280 = 137,280\text{ feet}\].

By doing this, you increase your comprehension of distance conversions and ensure that you can accurately work with different types of measurements.

Yards to Feet Conversion

The process of converting yards to feet is simpler than miles because not as many feet make up a yard. Each yard is equivalent to 3 feet.
This specific conversion is widely used, especially in sports and everyday projects that deal with more manageable distances compared to miles.

• Begin with the amount of yards you need to convert, such as 385 yards in our example.
• To perform the conversion, multiply the number of yards by 3:
  \[385 \times 3 = 1,155\text{ feet}\]

Understanding this step assures that you can easily change yard measurements into feet, helping you better navigate tasks involving shorter distances.

Algebraic Formula Application

Using algebraic formulas can assist whenever you need to manipulate numerical data systematically and accurately. In our marathon distance problem, we employ the formula \(f(m, y) = 5280m + 3y\) to convert miles and yards into feet.
This specific formula combines both miles and yards by transforming each into feet, then summing them for the total distance in feet.

• First, identify given values for miles and yards, as outlined in the problem: \( m = 26\) and \( y = 385\).
• Insert these values into the formula, so it becomes:
  \[f(26, 385) = 5,280 \times 26 + 3 \times 385\]
• Then, calculate each part separately, as shown in previous sections, and add the two results:
  \[137,280 + 1,155 = 138,435\]

Remember, applying formulas like this helps simplify calculations and ensures accurate results when dealing with combined units of measure.