Problem 79 - Elementary Algebra

Question

The Hubble Space Telescope is in low earth orbit traveling at an average speed of 16,950 miles per hour. What distance does it travel in \(11 / 2\) hours?

Step-by-Step Solution

Answer

The Hubble Space Telescope travels 93,225 miles in 5.5 hours.

Step 1: Identify the Formula

To find the distance traveled, we need to use the formula for distance, which is given by: \[ \text{Distance} = \text{Speed} \times \text{Time} \]

Step 2: Convert Time Fraction to Decimal

The time given is \(11/2\) hours. First, we convert this fraction to a decimal. \[ \frac{11}{2} = 5.5 \text{ hours} \]

Step 3: Substitute Values into the Formula

Using the formula from Step 1, substitute the speed \(16,950\) miles per hour and the time \(5.5\) hours: \[ \text{Distance} = 16,950 \times 5.5 \]

Step 4: Calculate the Distance

Perform the multiplication to find the distance:\[ 16,950 \times 5.5 = 93,225 \] So, the distance traveled is 93,225 miles.

Key Concepts

Speed and Distance

The concepts of speed and distance are tightly connected in physics and everyday life. Understanding them helps in calculating how far an object travels over time. Speed is the rate at which an object covers distance. It’s usually measured in miles per hour (mph) or kilometers per hour (kph). To find the distance an object travels, we use a simple and crucial formula:

• Distance = Speed × Time

This means that the distance is the product of how fast an object travels (its speed) and the time it takes. Let’s look at an example. If a telescope travels at 16,950 mph and you know the time it is traveling, you can calculate the distance covered by plugging these numbers into the formula. This is exactly what’s done with the Hubble Space Telescope example; we use the speed and the given time to find out how far it travels.

Fraction to Decimal Conversion

Converting fractions to decimals is a useful skill across different areas of math and science. It’s especially important when dealing with equations that involve decimals such as the distance formula. Converting fractions to decimals can simplify calculations and make them more manageable.To convert a fraction like \(\frac{11}{2}\) into a decimal, divide the numerator by the denominator:

• 11 (numerator) ÷ 2 (denominator) = 5.5

This means that \(\frac{11}{2}\) hours is equal to 5.5 hours. This conversion helps integrate with processes or formulas that require decimal inputs, like multiplying speed by time to figure out a traveled distance.

Multiplication in Algebra

Multiplication in algebra often involves more than just numbers; it can include variables, constants, and coefficients. However, even when multiplying plain numbers, it's crucial to understand how this operation fits into formulas and equations. In the given exercise, we execute multiplication to find distance:

• Distance = Speed (16,950) multiplied by Time (5.5)
• The product is 93,225

Here, the multiplication allows us to determine how far the telescope travels in a given time period. This is a practical example of using multiplication to solve real-world problems. Grasping multiplication in algebra is fundamental, as it forms the backbone of solving equations and manipulating algebraic expressions.