Problem 45 - Algebra 1: Concepts and Skills

Question

Use the following information. You are riding your bike to a pond that is 8 miles away. You have a choice to ride in the woods, on the road, or both. In the woods, you can ride at a speed of 10 mi/h. On the road, you can ride at a speed of 20 mi/h. Evaluate the expression for total time at 2 mile intervals.

Step-by-Step Solution

Answer

For 2 miles: In woods - 0.2 hours, On road - 0.1 hours. For 4 miles: In woods - 0.4 hours, On road - 0.2 hours. For 6 miles: In woods - 0.6 hours, On road - 0.3 hours. For 8 miles: In woods - 0.8 hours, On road - 0.4 hours.

Step 1: Calculate time to cover distance in the woods

To determine the time required to cover any distance in the woods, divide the distance by the travelling speed in the woods which is 10 mi/h. For instance, to cover 2 miles in the woods, the time would be \(\frac{2}{10}\) hour or 0.2 hours.

Step 2: Calculate time to cover distance on the road

The time to cover any distance on the road can be calculated similarly by dividing the distance by the speed on the road which is 20 mi/h. For instance, for 2 miles on the road, the time would be \(\frac{2}{20}\) hours or 0.1 hours.

Step 3: Repeat calculations for 2 mile intervals

Repeat above calculations for distances set at 2, 4, 6, and 8 miles. Remember, the time required changes depending on whether the distance is covered through the woods or on the road.

Key Concepts

Speed and Velocity

In distance-time problems like this exercise, understanding speed and velocity is key. Speed refers to how fast an object is moving regardless of its direction, while velocity includes both the speed and the direction of the object. For instance, if you are riding your bike to a pond, your speed might be 10 mi/h in the woods or 20 mi/h on the road. These speed values tell you how quickly you can travel in different terrains.

More precisely, speed is the distance an object covers per unit of time. In this exercise, the distance is measured in miles and the time is in hours, so speed is in miles per hour (mi/h). Velocity, on the other hand, would include information about whether you're traveling north, south, east, or west, but this problem doesn’t focus on direction, just on the speed values for different routes.

• Woods Speed: 10 mi/h
• Road Speed: 20 mi/h

Understanding these speeds allows us to calculate how long it will take to reach the pond depending on the route chosen and the hypothetical distance traveled for each step of the exercise.

Algebraic Expressions

Algebraic expressions are used in this problem to calculate time, which is a key component of many distance-time problems. The expression for time when riding a bike is determined by the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] In this exercise, we evaluate the expression at 2-mile intervals. To find out how long it would take to ride 2 miles in the woods, the expression becomes \( \frac{2}{10} \) hours, which equals 0.2 hours, as the speed in the woods is 10 mi/h. Similarly, to find out how long it would take on the road, it becomes \( \frac{2}{20} \) hours, which equals 0.1 hours, since the speed on the road is 20 mi/h.

**Evaluating Expressions**
Evaluating an expression involves substituting given values into the formula and computing the result. This is how we determine the time taken for different distances.

Here’s how it works:

• 2 Miles in Woods: \( \text{Time} = \frac{2}{10} = 0.2 \) hours
• 2 Miles on Road: \( \text{Time} = \frac{2}{20} = 0.1 \) hours

Unit Conversion

Unit conversion is an essential skill when dealing with distance-time problems, especially for converting time to different units. While this exercise uses miles and hours, you may often need to convert units to solve other similar problems.

**Distance and Speed Conversions**
When the units don’t match, it’s necessary to convert everything to ensure consistent and accurate calculations. Here, we are working with miles per hour, so there is no need to convert distances or speeds in this particular problem.

**Time Conversion**
For these calculations, the result is in hours. However, you might need to convert these to minutes or seconds, depending on the requirements of a specific problem. As an additional practice:

• 1 hour = 60 minutes
• 1 minute = 60 seconds

To convert the times we calculated earlier, you could use these conversions. For instance, 0.2 hours is equivalent to 12 minutes (0.2 × 60), and 0.1 hours is equivalent to 6 minutes (0.1 × 60). Such conversions make the understanding of time clearer and help in real-life applications.