Problem 34 - Algebra A Combined Function

Question

In the United States, a notable hang glider flight was a 303 -mile, \(8 \frac{1}{2}\) -hour flight from New Mexico to Kansas. What was the average rate during this flight?

Step-by-Step Solution

Answer

The average rate was approximately 35.65 miles per hour.

Step 1: Identify the given values

The problem provides the total distance traveled by the hang glider, which is 303 miles, and the total time taken for the flight, which is an 8 and a half hour or \(8 \frac{1}{2}\) hours.

Step 2: Convert mixed number to improper fraction

To simplify calculations, convert \(8 \frac{1}{2}\) hours into an improper fraction. The whole number 8 can be expressed as \(\frac{16}{2}\), so \(8 \frac{1}{2} = \frac{16}{2} + \frac{1}{2} = \frac{17}{2}\) hours.

Step 3: Use the average rate formula

The average rate can be calculated using the formula \(\text{Average Rate} = \frac{\text{Total Distance}}{\text{Total Time}}\). Here, that becomes:\[\text{Average Rate} = \frac{303\text{ miles}}{\frac{17}{2}\text{ hours}}\]

Step 4: Simplify the division

To divide by a fraction, multiply by its reciprocal. Thus, \[\text{Average Rate} = 303 \times \frac{2}{17}\]

Step 5: Calculate the average rate

Perform the multiplication:\[\text{Average Rate} = \frac{303 \times 2}{17} = \frac{606}{17}\approx 35.65\text{ miles per hour}\]

Key Concepts

Average Rate

Calculating the average rate is a common problem in word problems involving distance, time, and speed. It's essential to know how to find the average rate when you have the total distance and the total time. The formula for average rate is:\[\text{Average Rate} = \frac{\text{Total Distance}}{\text{Total Time}}\]This formula helps us understand how fast something is moving, on average, over a certain period. It is obtained by dividing the total distance traveled by the total time the journey took. In situations like the hang glider problem, it's important to have accurate values of distance and time, converted into the correct units, to calculate the average rate efficiently.
Keep in mind that average rate gives you a general idea of speed, not how speed varied during different moments of travel.

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are often used in everyday situations, like cooking or measuring, where values are not always whole numbers. For example, in the problem where the flight took \(8 \frac{1}{2}\) hours, \(8\) is the whole number and \(\frac{1}{2}\) is the fraction part.

• A mixed number can always be converted into an improper fraction. This conversion simplifies arithmetic operations, such as multiplication or division.
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator of the fraction. This sum becomes the new numerator, while the denominator remains the same.

So for \(8 \frac{1}{2}\), multiply \(8\times 2 = 16\), add \(1\), resulting in \(17\). Hence, \(8 \frac{1}{2} = \frac{17}{2}\).
Using improper fractions provides ease in calculations and prevents common errors when dealing with fractions.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator, such as \(\frac{17}{2}\). They are incredibly useful in mathematics because they simplify operations like multiplication, division, and comparison more than if performed with mixed numbers.

• They are called 'improper' because the value is greater than or equal to one, which is unlike 'proper' fractions that are less than one.
• Improper fractions can always be expressed as mixed numbers, but when performing calculations, sticking to improper fractions can avoid additional conversion steps and reduce complexity.

In the context of problems involving rates or complex fractions, improper fractions are preferable because they align easily with mathematical operations, making them ideal for division problems like our hang glider scenario.

Division of Fractions

Dividing fractions is a significant step when dealing with problems requiring the calculation of rates. It's easier than it might seem, but it requires knowing how to multiply by the reciprocal. For example, to divide \(\frac{303}{1}\) by \(\frac{17}{2}\), consider the reciprocal of \(\frac{17}{2}\), which is \(\frac{2}{17}\). Instead of division, you multiply:\[303 \times \frac{2}{17}\]

• Always remember: dividing by a fraction is synonymous with multiplying by its reciprocal.
• Reciprocal means flipping the fraction, i.e., making the numerator the denominator and vice versa.

This approach simplifies calculation and helps avoid mistakes, ensuring a smoother process to derive the answer, as in finding the average rate of the glider.