A Right Rectangular Prism Has A Base Area Of 2 1 2 2 \frac{1}{2} 2 2 1 ​ Square Inches And A Height Of 11 1 2 11 \frac{1}{2} 11 2 1 ​ Inches. What Is The Volume Of The Prism?A. 14 In 3 14 \text{ In}^3 14 In 3 B. 28 3 4 In 3 28 \frac{3}{4} \text{ In}^3 28 4 3 ​ In 3 C. $71

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Understanding the Problem

To find the volume of a right rectangular prism, we need to multiply the base area by the height. The base area is given as $2 \frac{1}{2}$ square inches, and the height is given as $11 \frac{1}{2}$ inches. We will first convert these mixed numbers to improper fractions to make the calculation easier.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and then add the numerator. The result is then written as the new numerator over the denominator.

• $2 \frac{1}{2}$ can be converted to an improper fraction as follows:
  • Whole number part: 2
  • Denominator: 2
  • Numerator: 1
  • Multiply whole number part and denominator: 2 x 2 = 4
  • Add numerator: 4 + 1 = 5
  • Result: $\frac{5}{2}$
• $11 \frac{1}{2}$ can be converted to an improper fraction as follows:
  • Whole number part: 11
  • Denominator: 2
  • Numerator: 1
  • Multiply whole number part and denominator: 11 x 2 = 22
  • Add numerator: 22 + 1 = 23
  • Result: $\frac{23}{2}$

Calculating the Volume

Now that we have the base area and height in improper fraction form, we can calculate the volume by multiplying them together.

Volume = base area x height = $\frac{5}{2}$ x $\frac{23}{2}$ = $\frac{5 x 23}{2 x 2}$ = $\frac{115}{4}$

Converting the Volume to a Mixed Number

To convert the improper fraction to a mixed number, we divide the numerator by the denominator and write the remainder as the new numerator.

$\frac{115}{4}$ can be converted to a mixed number as follows:

• Divide numerator by denominator: 115 ÷ 4 = 28 with a remainder of 3
• Result: $28 \frac{3}{4}$

Conclusion

The volume of the right rectangular prism is $28 \frac{3}{4}$ cubic inches.

Discussion

The problem requires us to find the volume of a right rectangular prism given its base area and height. We first convert the mixed numbers to improper fractions to make the calculation easier. Then, we multiply the base area and height together to find the volume. Finally, we convert the improper fraction to a mixed number to get the final answer.

Step-by-Step Solution

1. Convert the mixed numbers to improper fractions.
2. Multiply the base area and height together to find the volume.
3. Convert the improper fraction to a mixed number to get the final answer.

Common Mistakes

• Forgetting to convert the mixed numbers to improper fractions.
• Multiplying the base area and height together incorrectly.
• Forgetting to convert the improper fraction to a mixed number.

Real-World Applications

Finding the volume of a right rectangular prism has many real-world applications, such as:

• Calculating the volume of a box or container.
• Finding the volume of a tank or reservoir.
• Determining the volume of a building or structure.

Practice Problems

1. A right rectangular prism has a base area of $3 \frac{1}{3}$ square inches and a height of $12 \frac{1}{3}$ inches. What is the volume of the prism?
2. A right rectangular prism has a base area of $4 \frac{1}{4}$ square inches and a height of $10 \frac{1}{4}$ inches. What is the volume of the prism?

Solutions to Practice Problems

1. Convert the mixed numbers to improper fractions: $\frac{10}{3}$ and $\frac{37}{3}$

2. Multiply the base area and height together: $\frac{10}{3} x \frac{37}{3} = \frac{370}{9}$

3. Convert the improper fraction to a mixed number: $41 \frac{1}{9}$

4. Convert the mixed numbers to improper fractions: $\frac{17}{4}$ and $\frac{41}{4}$

5. Multiply the base area and height together: $\frac{17}{4} x \frac{41}{4} = \frac{697}{16}$

6. Convert the improper fraction to a mixed number: $43 \frac{13}{16}$
   

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Q&A: Volume of a Right Rectangular Prism

Q: What is the formula for finding the volume of a right rectangular prism?

A: The formula for finding the volume of a right rectangular prism is: Volume = base area x height.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number part by the denominator and then add the numerator. The result is then written as the new numerator over the denominator.

Q: What is the base area of the prism in the problem?

A: The base area of the prism is $2 \frac{1}{2}$ square inches.

Q: What is the height of the prism in the problem?

A: The height of the prism is $11 \frac{1}{2}$ inches.

Q: How do I calculate the volume of the prism?

A: To calculate the volume of the prism, you multiply the base area by the height. In this case, you would multiply $\frac{5}{2}$ by $\frac{23}{2}$.

Q: What is the volume of the prism in the problem?

A: The volume of the prism is $28 \frac{3}{4}$ cubic inches.

Q: What are some real-world applications of finding the volume of a right rectangular prism?

A: Finding the volume of a right rectangular prism has many real-world applications, such as calculating the volume of a box or container, finding the volume of a tank or reservoir, and determining the volume of a building or structure.

Q: What are some common mistakes to avoid when finding the volume of a right rectangular prism?

A: Some common mistakes to avoid when finding the volume of a right rectangular prism include forgetting to convert the mixed numbers to improper fractions, multiplying the base area and height together incorrectly, and forgetting to convert the improper fraction to a mixed number.

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, you divide the numerator by the denominator and write the remainder as the new numerator.

Q: What is the final answer to the problem?

A: The final answer to the problem is $28 \frac{3}{4}$ cubic inches.

Q: Can you provide some practice problems for finding the volume of a right rectangular prism?

A: Here are a few practice problems:

1. A right rectangular prism has a base area of $3 \frac{1}{3}$ square inches and a height of $12 \frac{1}{3}$ inches. What is the volume of the prism?
2. A right rectangular prism has a base area of $4 \frac{1}{4}$ square inches and a height of $10 \frac{1}{4}$ inches. What is the volume of the prism?

Q: Can you provide the solutions to the practice problems?

A: Here are the solutions to the practice problems:

1. Convert the mixed numbers to improper fractions: $\frac{10}{3}$ and $\frac{37}{3}$
2. Multiply the base area and height together: $\frac{10}{3} x \frac{37}{3} = \frac{370}{9}$
3. Convert the improper fraction to a mixed number: $41 \frac{1}{9}$
4. Convert the mixed numbers to improper fractions: $\frac{17}{4}$ and $\frac{41}{4}$
5. Multiply the base area and height together: $\frac{17}{4} x \frac{41}{4} = \frac{697}{16}$
6. Convert the improper fraction to a mixed number: $43 \frac{13}{16}$