Problem 85 - Prealgebra

Question

Aerostar International makes a hot air balloon called the Rally 105 that has a volume of 105,400 cubic feet. Another balloon, the Rally \(126,\) was designed with a volume that is approximately \(\frac{6}{5}\) the volume of the Rally \(105 .\) Find the volume of the Rally 126 to the nearest hundred cubic feet.

Step-by-Step Solution

Answer

The volume of the Rally 126 balloon is 126,500 cubic feet.

Step 1: Understanding the Problem

We are given the volume of the Rally 105 balloon as 105,400 cubic feet. We need to find the volume of the Rally 126 balloon, knowing it is designed with a volume that is \(\frac{6}{5}\) times the volume of the Rally 105.

Step 2: Setting Up the Equation

To find the volume of the Rally 126, we set up the equation based on the given ratio: \(V_{126} = \frac{6}{5} \times V_{105}\), where \(V_{105} = 105,400\) cubic feet.

Step 3: Calculating the Volume

Substitute the value of \(V_{105}\) into the equation: \(V_{126} = \frac{6}{5} \times 105,400\).

Step 4: Performing the Multiplication

Calculate \(V_{126} = 1.2 \times 105,400\) to get the exact volume of the Rally 126. This simplifies to \(V_{126} = 126,480\) cubic feet.

Step 5: Rounding the Result

Round 126,480 to the nearest hundred cubic feet, which results in 126,500 cubic feet.

Key Concepts

Ratios in Volume Calculation

Ratios are a fundamental tool in mathematical calculations, especially in cases where quantities are related to each other. In the context of volume calculation, ratios help us compare and discern the size relationships between different objects. In our problem, the Rally 126 balloon's volume is given in terms of a ratio to the Rally 105 balloon. Specifically, the Rally 126 is designed to have a volume that is \( \frac{6}{5} \) times that of the Rally 105. The fraction \( \frac{6}{5} \) indicates that the Rally 126 is one-step larger by this proportional amount. It means for every 5 cubic feet of the Rally 105, the Rally 126 has 6 cubic feet. Understanding how to apply ratios is crucial in correctly setting up equations for solving volume problems. They help identify how quantities scale relative to each other, making it easier to calculate the desired volume by applying simple arithmetic rules.

The Role of Multiplication in Volume Calculations

Multiplication is an essential operation when calculating volumes, especially when using ratios to find a specific value. In the given exercise, multiplication helps us translate the ratio into a numerical quantity that represents the actual volume of the Rally 126 balloon. Here's how it works:

• Identify the volume of the first object, which is the Rally 105 balloon with a volume of 105,400 cubic feet.
• Determine the given ratio, in this case, \( \frac{6}{5} \), that extends the volume of the first balloon to find the second.
• Multiply: Convert the fraction \( \frac{6}{5} \) to a decimal, which is 1.2, then multiply 1.2 by the volume of Rally 105.

This multiplication process gives us 126,480 cubic feet, the exact volume of the Rally 126 balloon before rounding. Multiplication allows for precise calculations, particularly when working with larger numbers, making it a vital tool in handling problems involving volume ratios.

Rounding Numbers for Practical Use

Rounding numbers is an important step in presenting calculations, especially when dealing with volumes, where precision could be necessary, yet practical values are often preferred. Rounding is the process of simplifying a number to make it easier to work with or present in space-constrained contexts. For instance, in the volume calculation for the Rally 126 balloon, the exact result from multiplication was 126,480 cubic feet. However, we are interested in showing this value to the nearest hundred cubic feet. Here’s a very brief on how to round to the nearest hundred:

• Inspect the tens digit: For 126,480, this digit is 8.
• If this digit is 5 or more, increase the hundreds digit by one.
• If it is less, leave the hundreds digit as it is and zero out all digits following the hundreds place.

Thus, 126,480 rounds up to 126,500 cubic feet. This approach makes numbers manageable and easier to communicate, without significantly affecting the precision required for engineering calculations.