Question

5. a) Identify 10 different rectangular prisms with whole-number dimensions that have a volume of 144 cm³. Show your work.

b) The volume of a rectangular prism with whole-number dimensions has a volume that is a prime number. What are its dimensions? Use an example to support your explanation.

Answer 1

a) Ten different rectangular prisms with whole-number dimensions that have a volume of 144 cm³ are:

1.
`\( 1 * 1 * 144 \)` cm³

2.
`\( 1 * 2 * 72 \)` cm³

3.
`\( 1 * 3 * 48 \)` cm³

4.
`\( 1 * 4 * 36 \)` cm³

5.
`\( 1 * 6 * 24 \)` cm³

6.
`\( 1 * 8 * 18 \)` cm³

7.
`\( 1 * 9 * 16 \)` cm³

8.
`\( 1 * 12 * 12 \)` cm³

9.
`\( 2 * 2 * 36 \)` cm³

10.
`\( 2 * 3 * 24 \)` cm³

b) The dimensions of a rectangular prism with whole-number dimensions and a volume that is a prime number could be
`\( 1 * 1 * 144 \)` cm³. For example, a rectangular prism with dimensions
`\( 1 * 1 * 144 \)`cm³ has a volume of 144 cm³, which is not a prime number.

Step-by-step explanation:

Certainly! Let's go through the detailed calculations for both parts of the question.

a) Identifying Rectangular Prisms with a Volume of 144 cm³:

To find rectangular prisms with a volume of 144 cm³ and whole-number dimensions, we need to consider the factors of 144 and arrange them into sets of three.

Factors of 144:

1.
`\(1 * 1 * 144\) cm^3`

2.
`\(1 * 2 * 72\) cm^3`

3.
`\(1 * 3 * 48\) cm^3`

4.
`\(1 * 4 * 36\) cm^3`

5.
`\(1 * 6 * 24\) cm^3`

6.
`\(1 * 8 * 18\) cm^3`

7.
`\(1 * 9 * 16\) cm^3`

8.
`\(1 * 12 * 12\) cm^3`

9.
`\(2 * 2 * 36\) cm^3`

10.
`\(2 * 3 * 24\) cm^3`

These combinations result in rectangular prisms with whole-number dimensions and a volume of 144 cm³.

b) Rectangular Prism with Whole-Number Dimensions and a Prime Volume:

The goal here is to find dimensions that result in a rectangular prism with a volume that is a prime number.

Incorrect example:
`\(1 * 1 * 144\)` cm³

Correct example:
`\(2 * 2 * 36\)` cm³

For the correct example, the calculations are as follows:

`\[ \text{Volume} = \text{Length} * \text{Width} * \text{Height} \]`

`\[ \text{Volume} = 2 * 2 * 36 = 144 \, \text{cm³} \]`

In this case, the volume is 144 cm³, which is not a prime number. Let me correct the example:

Corrected example:
`\(2 * 2 * 36\)` cm³

Now, the volume is 144 cm³, and the dimensions are whole numbers, but the volume is not a prime number. I apologize for the confusion in my previous responses.

Answer 2

Ten different rectangular prisms with a volume of 144 cm³ are identified. It is explained that a rectangular prism with a volume that is a prime number cannot have whole-number dimensions since prime numbers only have two distinct factors.

Step-by-step explanation:

The question involves finding different rectangular prisms with a specific volume and understanding the concept of volume in terms of whole numbers and prime numbers. We're asked to identify 10 different rectangular prisms with whole-number dimensions that have a volume of 144 cm³ and then consider the possibility of a prism with a prime number volume.

• (1, 144, 1)
• (2, 72, 1)
• (3, 48, 1)
• (4, 36, 1)
• (6, 24, 1)
• (8, 18, 1)
• (9, 16, 1)
• (12, 12, 1)
• (3, 8, 6)
• (4, 4, 9)

For part b, a rectangular prism with dimensions that yield a prime number volume is not possible since a volume is computed by multiplying the length, width, and height, and a prime number has only two distinct divisors: 1 and itself. Therefore, any non-unit dimensions multiplied together will result in a non-prime product.

An example supporting this explanation: Consider a prime number like 5. We cannot find whole number dimensions for a rectangular prism that multiply to 5 because this would require one of the factors to be 1 and the other two factors to be 5, which is not a whole number.