Question

In 40 seconds, a mountain biker travels 230 feet. What is her speed in miles per hour? (Note: 1 mile \(=5280\) feet) $$ \begin{array}{r} \frac{230 \times 60 \times 60}{40 \times 5280} \\ \frac{230 \times 5280}{40 \times 60 \times 60} \\ \frac{40 \times 60 \times 60}{230 \times 5280} \\ \text { I don't know. } \end{array} $$

Short answer

The correct option is \(\frac{230 \times 60 \times 60}{40 \times 5280}\). It represents the speed of the mountain biker in miles per hour.

Step-by-step solution

Understanding the conversion ratios

Recognize that 1 mile equals 5280 feet. Thus, to convert feet to miles, you can divide the number of feet by 5280. Similarly, there are 60 seconds in a minute and 60 minutes in an hour, so to convert seconds to hours, you would divide the number of seconds by 3600 (60*60).

Convert distance from feet to miles

The mountain biker travels 230 feet, which needs to be converted into miles by dividing by 5280. Hence, the distance in miles can be calculated as: \[ Distance = \frac{230}{5280} \] miles.

Convert time from seconds to hours

The biker has taken 40 seconds. To convert this into hours, you divide by 3600. This gives: \[ Time = \frac{40}{3600} \] hours.

Calculate speed in miles per hour

Speed is calculated as distance divided by time. Therefore, the speed of the mountain biker in miles per hour is: \[ Speed = \frac{Distance}{Time} = \frac{\frac{230}{5280}}{\frac{40}{3600}} \]

Simplify the expression

Simplify the fraction to get the speed of the mountain biker in miles per hour. This will be the final answer.

Key concepts

Distance Conversion

Distance conversion is an essential mathematical skill, especially when dealing with different units of measurement, like moving from feet to miles. Given that 1 mile equals 5280 feet, this conversion becomes straightforward by using division. For instance, if you have a distance measured in feet, to express it in miles, you simply divide the total number of feet by 5280. This gives you a clear understanding of the distance traveled in miles.

In our exercise, the mountain biker covers a distance of 230 feet. To convert this distance from feet to miles, apply the conversion factor:

• Distance in miles = \( \frac{230}{5280} \)

Thus, this fraction represents the distance in miles that the biker has traveled. Converting different units into a common unit of measurement, like miles, is essential to perform further computations like speed.

Time Conversion

Time conversion involves changing a time measurement from one unit to another, for example from seconds to hours. This is particularly practical when dealing with formulas in physics or everyday computations that require consistency in units. The number of seconds in an hour is 3600 since there are 60 seconds in a minute and 60 minutes in an hour.

If you're given a time in seconds and need it in hours, you divide by 3600:

• Time in hours = \( \frac{40}{3600} \)

So for the biker taking 40 seconds, this formula lets us find out how long that is in hours. Consistent time units are crucial when calculating speed to ensure that your results are accurate and meaningful across applications.

Speed Calculation

Speed calculation is determining how fast an object moves over a certain distance in a specific amount of time. It is typically represented by the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). This formula helps break down the measurement of motion into simple, understandable components.

Using the previous conversions:

• Distance in miles = \( \frac{230}{5280} \)
• Time in hours = \( \frac{40}{3600} \)

You can calculate speed by putting these together:

• Speed = \( \frac{\frac{230}{5280}}{\frac{40}{3600}} \)

This calculation gives the speed in miles per hour. Understanding how to use the speed formula effectively allows one to analyze and compare different movements and dynamics.

Miles per Hour

Miles per hour (mph) is a unit of speed commonly used in many countries to describe the speed of vehicles and other moving objects. It tells us how many miles an object can travel in one hour, making it an intuitive measurement for daily use.

After calculating speed in miles per hour using the formula mentioned earlier, we achieve an understanding of how fast the mountain biker travels relative to a standard speed unit.

In our example, the simplified fraction from the speed calculation reflects this unit, offering a universal way to communicate speed that's easily comparable across different contexts. Seeing speed as miles per hour assists in visualizing how distances are covered over time, enabling predictions and planning based on velocity.