Problem 40 - Algebra A Combined Function

Question

A family is planning their vacation to Disney World. They will drive from a small town outside New Orleans, Louisiana, to Orlando, Florida, a distance of 700 miles. They plan to average a rate of \(55 \mathrm{mph}\). How long will this trip take?

Step-by-Step Solution

Answer

The trip will take approximately 12 hours and 44 minutes.

Step 1: Identify the Problem Type

This problem involves calculating the time for a trip when distance and speed are known. It requires the use of the distance-speed-time relationship formula.

Step 2: Recall the Formula

The formula to find time when distance and speed are known is given by: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

Step 3: Substitute Known Values into the Formula

Substitute 700 miles for distance and 55 mph for speed into the formula: \[ \text{Time} = \frac{700}{55} \]

Step 4: Perform the Calculation

Divide 700 by 55 to find the time: \[ \text{Time} \approx 12.727 \] hours.

Step 5: Round to a Practical Time

Since time is usually expressed in practical units, round 12.727 to 12.73 hours or approximately 12 hours and 44 minutes.

Key Concepts

Distance-Speed-Time Relationship

When planning a trip, one of the key calculations you'll need to make is how long the journey will take. This involves understanding the distance, speed, and time relationship. These three things are connected by the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] To use this formula, consider the distance to your destination and the average speed at which you'll be traveling. By dividing the total distance by the speed, you can find the total time your trip will take.
Imagine you're planning a drive of 700 miles and intend to travel at an average speed of 55 mph. Using the formula, the time taken to complete the trip would be \( \frac{700}{55} \), which equals approximately 12.727 hours.
This simple mathematical relationship helps in creating accurate travel plans and can be adjusted to accommodate changes in speed or distance.

• The greater your speed, the less time the journey takes.
• For longer distances, travels will naturally take more time at the same speed.

Using this formula is fundamental in various real-life scenarios, like road trips, airline travel, and even planning your daily commute.

Problem Solving in Mathematics

Mathematical problem solving is a methodical approach to finding a solution to a question. Let's use our travel problem as a case study to break down the steps required.
Firstly, **identify the problem.** In this case, we need to determine the travel time.
Next, **recall relevant information** or formulas that can help, such as the distance-speed-time formula. Once you know what you're working with, **substitute the known values** into the formula. Insert 700 miles for distance and 55 mph for speed:
\[ \text{Time} = \frac{700}{55} \] Now, **perform calculations** to find your answer, which we did and found the answer as approximately 12.727 hours. Finally, when coming to the result, it may require additional steps like rounding for practical usage, which brings clarity to the context. This structured problem-solving approach is universal in mathematics, aiding in breaking down complex problems into manageable steps.

Rounding Numbers

In mathematics, rounding is a neat skill that helps create more meaningful results. After calculating the time to be around 12.727 hours, this long decimal isn't practical for most real-world uses.
To round numbers, you usually look at what decimal place you want to round to. Often, rounding is done to two decimal places when dealing with time or currency. Let's transform 12.727 hours to a simpler form.
First, look at the third decimal place (7 in this instance), which will affect rounding the second decimal place up or down. For instance:

• If the third digit is 5 or more, increase the second-by-one. It turns 12.727 into 12.73.
• If less than 5, leave the second digit as it is.

Moreover, for a clearer concept like time, transforming 12.73 hours to hours and minutes results in approximately 12 hours and 44 minutes.
Rounding not only simplifies numbers but makes them far more digestible in both written and conversational formats.