Problem 64 - Algebra A Combined Function

Question

On a road course, a CART car's speed can average up to around 105 mph. Based on this speed, how long would it take a CART driver to travel from Los Angeles to New York City, a distance of about 2810 miles by road, without stopping? Round to the nearest tenth of an hour.

Step-by-Step Solution

Answer

It would take approximately 26.8 hours to travel the distance.

Step 1: Identify Variables and Known Information

We have the average speed of the CART car which is 105 mph, and the distance from Los Angeles to New York City which is 2810 miles.

Step 2: Use the Formula for Time

To find the time taken to travel a certain distance, we can use the formula: \[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]

Step 3: Substitute the Known Values

Substitute the known values into the formula: \[\text{Time} = \frac{2810 \text{ miles}}{105 \text{ mph}}\]

Step 4: Calculate the Time

Perform the division to calculate the time:\[\text{Time} = \frac{2810}{105} \approx 26.8 \text{ hours}\]

Step 5: Round to the Nearest Tenth

The calculated time is approximately 26.8 hours, which is already rounded to the nearest tenth.

Key Concepts

Speed and Distance Problems

Understanding speed and distance problems is a vital skill in algebra. These problems often require you to determine one of the three key elements: speed, distance, or time. The relationship between these elements is given by the formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \). This means to find any of the other two elements, you can rearrange this equation. For example, if you need to find the time it takes for an object to travel a certain distance at a given speed, the formula becomes \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). The key to solving these problems is first to identify what you know and what you need to find. Once you have these, substitute them into the formula to determine the unknown variable. This straightforward approach makes these problems manageable with practice.

Average Speed Calculation

Calculating average speed is useful when you need to determine how fast something is moving over a period of time. Average speed is determined by the total distance traveled divided by the total time taken to travel that distance. This is expressed mathematically as \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \). This calculation provides a single speed that represents the entire journey, even if the actual speed varied throughout. It's crucial in situations, such as traveling long distances, where maintaining a constant speed isn't possible. Calculating average speed enables you to plan effectively, ensuring you know approximately how long a trip will take given the overall conditions.

Time Calculation

Time calculation plays a significant role in solving real-world problems involving motion. Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), you can find how long it takes to travel a certain distance at a given speed. For instance, to calculate the time a car takes to travel from Los Angeles to New York City at an average speed of 105 mph, substitute the known values into the formula: \( \text{Time} = \frac{2810 \ \text{miles}}{105 \ \text{mph}} \). Simplifying this gives approximately 26.8 hours. Determining time is critical in planning trips and ensuring timely arrivals. Always remember to keep track of your units, ensuring that you don't mix miles with kilometers or hours with minutes unless converting appropriately.

Rounding Numbers

Rounding numbers is a useful technique that simplifies complex figures by approximating them to a specific decimal place. In scientific and practical calculations, rounding allows for more concise answers. When rounding a number to the nearest tenth, you look at the hundredths place to decide whether to round up or down. If the digit in the hundredth spot is 5 or more, round up the tenth place digit; if it's less than 5, keep the tenth place digit the same. For example, the number 26.754 rounds to 26.8 because the "5" in the hundredths place signals an increase in the tenth digit. Rounding is particularly important in scenarios that require a level of precision, such as estimating travel times, like in our CART driver example.