Question

Exercises \(57-59\), you are going on a trip to the Natural History Museum. At 9:00 A.M., you leave for the museum, which is 120 miles away. At 10: 15 A.M., you are 63 miles away from the museum. Find the distance you are from the museum after you have traveled 2 hours.

Short answer

After two hours of travel, you will be 28.8 miles away from the museum.

Step-by-step solution

Calculate the Speed of the Trip

To determine the speed, take the distance covered and divide it by the time taken. This is given as (120 miles - 63 miles) / (10:15 A.M - 9:00 A.M). The distance covered in this time span is 57 miles. Convert the time difference to hours for consistency in units (1 hour and 15 minutes is equal to 1.25 hours). So the speed would be \( \frac{57}{1.25} = 45.6 \) mph.

Use Speed to Determine the Distance After Two Hours

With the speed figured out, the distance covered after two hours can be determined by multiplying the speed by time (distance = speed*time). Hence, the distance covered after two hours would be \( 45.6 mph * 2 hours = 91.2 \) miles.

Calculate the Distance from the Museum

To find the remaining distance to the museum, subtract the distance covered from the total distance from the starting point to the museum. This gives \( 120 miles - 91.2 miles = 28.8 \) miles.

Key concepts

Calculating Speed

Understanding how to calculate speed is vital for solving problems involving motion. Speed defines how fast an object is moving and is commonly represented as a ratio of distance traveled to the time taken to travel that distance. In essence, speed is distance per time.

To calculate speed, you can use the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
For the exercise given, you traveled a distance of 120 miles to the museum and found yourself 63 miles away after 1 hour and 15 minutes, or 1.25 hours. Using the distance formula for speed, the calculation was: \( 45.6 \text{ mph} = \frac{57 \text{ miles}}{1.25 \text{ hours}} \). Here are some tips to make sure you get it right:

• Always express time in the same units, here hours are used.
• Ensure the distance units are consistent, miles in this case.
• Use precise values to convert mixed time units (like hours and minutes) into a single unit.

Once the speed is known, it can be applied to find distances or times for other parts of a trip.

Time and Distance Conversion

Converting time and distance from one set of units to another is a common task in algebra problems, particularly when dealing with speed and travel. For time conversions, understanding that 1 hour is equal to 60 minutes is essential.

In the context of the museum exercise, the initial time was given as 9:00 A.M. and the next time check was at 10:15 A.M. This time interval needs to be converted into hours as speed is often measured in miles per hour (mph). To convert 1 hour and 15 minutes to just hours, you'd do the following: \[ 1 \text{ hour} + \frac{15 \text{ minutes}}{60} = 1.25 \text{ hours} \].
Similar conversions can be made for distance if needed, though in this exercise, the distance was already given in miles. Keep in mind, converting units correctly ensures that the speed calculation is accurate and that any further computations based on speed are reliable.

Distance-Speed-Time Relationship

The relationship between distance, speed, and time is a cornerstone of motion problems in algebra. The formula to find any of the three if the other two are known is straightforward:
\[ \text{Distance} = \text{Speed} \times \text{Time} \].

Applying this formula to our museum trip exercise, we can determine the distance traveled after any amount of time, provided we know the speed. For example, after two hours, you can calculate the distance covered. With a speed of 45.6 mph, for two hours of travel, the distance is calculated as \( \text{Distance} = 45.6 \text{ mph} \times 2 \text{ hours} = 91.2 \text{ miles} \).
This shows that the distance from the starting point is 91.2 miles, and the remaining distance to the museum is then \( 120 \text{ miles} - 91.2 \text{ miles} = 28.8 \text{ miles} \) away.
Remember, this relationship holds true only if the speed remains constant. If the speed changes during the trip, you would need to calculate the distance using the speed for each segment of the trip separately.