Problem 33 - Algebra A Combined Function

Question

The \(\mathrm{X}-30\) is a "space plane" that skims the edge of space at 4000 miles per hour. Neglecting altitude, if the circumference of Earth is approximately 25,000 miles, how long will it take for the \(\mathrm{X}-30\) to travel around Earth?

Step-by-Step Solution

Answer

It takes 6.25 hours for the X-30 to travel around Earth.

Step 1: Understanding the Problem

We need to find out how long it will take for the X-30 to travel around the Earth at a constant speed of 4000 miles per hour, given that the circumference of the Earth is approximately 25000 miles.

Step 2: Identify the Formula Needed

Since time is equal to distance divided by speed, we can use the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] to calculate the travel time of the space plane.

Step 3: Calculate the Time

Substitute the given values into the formula: \[ \text{Time} = \frac{25000 \text{ miles}}{4000 \text{ miles per hour}} = 6.25 \text{ hours} \]. Therefore, it takes 6.25 hours for the X-30 to travel around the Earth.

Key Concepts

Circumference

The circumference is a crucial aspect when dealing with situations that involve circular motion or travel around a spherical object, like Earth. Simply put, circumference is the distance around a circle. For planets, which are roughly spherical, the circumference is the total distance one would travel if they walked around the entire planet along the equator. In our problem, the Earth's circumference is given as 25,000 miles. This value represents the complete loop around the planet at the equator. Understanding the concept of circumference helps in solving problems related to circular travels, like when calculating routes for satellites or, in this case, an advanced space plane skimming the edge of space. A common formula to remember is:

• For a circle: Circumference = \( 2 \pi r \), where \( r \) is the radius of the circle.

However, in many practical applications, we use predefined measurements like the Earth's circumference.

Speed-Distance-Time Relationship

This relationship is fundamental in understanding motion and involves three key quantities: speed, distance, and time. The formula that connects these quantities is:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]In essence, this formula allows us to calculate the time it takes for an object to travel a certain distance at a given speed. If you know any two of these quantities, you can use the formula to find the third. In our exercise, we know:

• Distance = 25,000 miles (the circumference of Earth)
• Speed = 4,000 miles per hour

By substituting these values into our formula, we solved for time:\[ \text{Time} = \frac{25000}{4000} = 6.25 \text{ hours} \]This calculation shows us how long it takes for the X-30 to make a full journey around Earth at its steady speed. Remembering this formula and understanding the relationships between these quantities aids in various real-world applications, including travel planning and engineering.

Problem-Solving

Learning effective problem-solving strategies is key to tackling mathematical exercises like this one. When faced with a problem, it's important to: 1. **Understand the Problem**: Grasp the question at hand. Here, we needed to determine the time for the X-30 to circle Earth.
2. **Identify Relevant Information**: This includes known values and what is required to be solved. We were given Earth's circumference and the plane's speed.
3. **Select the Appropriate Formula or Tool**: Often, knowing the right formula is half the battle. In this case, the speed-distance-time relationship was our go-to.
4. **Perform the Calculations**: Substitute the known values into the formula and solve. We found the time as 6.25 hours.
5. **Reflect on the Result**: Check if the solution is reasonable. Given the enormous speed of the X-30 and Earth's sizeable circumference, our calculated time should make logical sense. Practicing these steps enhances your problem-solving skills, not just in algebra or mathematics, but in day-to-day challenges as well.