Problem 85 - Algebra A Combined Function

Question

Flying fish do not actually fly, but glide. They have been known to travel a distance of 1300 feet at a rate of 20 miles per hour. How many seconds would it take to travel this distance?

Step-by-Step Solution

Answer

The flying fish would take approximately 44.316 seconds to travel 1300 feet at 20 mph.

Step 1: Convert Distance to Miles

First, we convert the distance the fish travels from feet to miles, as we have the speed in miles per hour. We know that 1 mile equals 5280 feet. So we divide the distance in feet by the number of feet in a mile: \[ 1300 \, \text{ft} \times \frac{1 \, \text{mile}}{5280 \, \text{ft}} = 0.2462 \, \text{miles} \]

Step 2: Calculate Time in Hours

Using the speed of the fish, which is 20 miles per hour, we calculate the time it takes to travel 0.2462 miles. Since speed is distance divided by time, we rearrange for time: \[ \text{Time (in hours)} = \frac{\text{Distance}}{\text{Speed}} = \frac{0.2462}{20} = 0.01231 \, \text{hours} \]

Step 3: Convert Time to Seconds

Finally, we convert the time from hours to seconds. There are 3600 seconds in an hour, so we multiply the time in hours by the number of seconds in an hour: \[ 0.01231 \, \text{hours} \times 3600 \, \text{seconds per hour} = 44.316 \, \text{seconds} \]

Key Concepts

Distance Conversion

When dealing with problems involving measurements, it's often necessary to convert units to make calculations easier. In our flying fish example, the distance given was 1300 feet, but the speed was provided in miles per hour. To use the speed to find out how long the fish glides, we need to convert feet into miles.

Here’s how you approach this kind of conversion:

• Understand that 1 mile equals 5280 feet.
• Use a conversion factor to change feet into miles. This is done by dividing the number of feet by the number of feet in one mile.

For the flying fish, the conversion was: \( 1300 \, \text{ft} \times \frac{1 \, \text{mile}}{5280 \, \text{ft}} = 0.2462 \, \text{miles} \).

Converting distances accurately is crucial in problems requiring different measurement units. Always remember to multiply or divide by the correct factor.

Speed Calculation

Speed is a measure of how fast something is moving and is typically calculated as the distance traveled per unit of time. In many practical scenarios, understanding and calculating speed is essential. In our flying fish problem, the speed is given as 20 miles per hour.

Here's how you calculate speed-related problems:

• Speed is defined as \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
• If you have to find the time, the formula rearranges to \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

For instance, the fish needs to travel 0.2462 miles at 20 miles per hour. Using the formula, the time in hours is computed as: \( \frac{0.2462}{20} = 0.01231 \, \text{hours} \).

Always ensure that when you are calculating time, distance, or speed, the units are consistent for all measures.

Time Conversion

Time conversion is often necessary to express time in useful ways, such as converting hours into minutes or seconds. In the case of our flying fish exercise, the calculated time in hours had to be converted to seconds to answer the problem correctly.

To convert from hours to seconds, consider the following:

• Know that one hour contains 3600 seconds.
• Multiply the number of hours by 3600 to find out the time in seconds.

For our problem, once we have determined that the fish traveled in 0.01231 hours, we convert this into seconds: \( 0.01231 \, \text{hours} \times 3600 \, \text{seconds per hour} = 44.316 \, \text{seconds} \).

Accurate time conversion is essential to give precise answers in problems involving time measurements.