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A rectangular prism has a length of [tex]$4.2 \, \text{cm}[tex]$[/tex], a width of [tex]$[/tex]5.8 \, \text{cm}[tex]$[/tex], and a height of [tex]$[/tex]9.6 \, \text{cm}[tex]$[/tex]. A similar prism has a length of [tex]$[/tex]14.7 \, \text{cm}[tex]$[/tex], a width of [tex]$[/tex]20.3 \, \text{cm}[tex]$[/tex], and a height of [tex]$[/tex]33.6 \, \text{cm}$[/tex].

The dimensions of the smaller prism are each multiplied by what factor to produce the corresponding dimensions of the larger prism?

A. [tex]3 \frac{1}{2}[/tex]
B. 4
C. [tex]4 \frac{5}{6}[/tex]
D. 5



Answer :

GinnyAnswer
To determine the factor by which each dimension of the smaller prism has been multiplied to obtain the dimensions of the larger prism, follow these steps:

1. Identify the dimensions of both prisms:
- Smaller prism: length \(4.2 \text{ cm}\), width \(5.8 \text{ cm}\), height \(9.6 \text{ cm}\).
- Larger prism: length \(14.7 \text{ cm}\), width \(20.3 \text{ cm}\), height \(33.6 \text{ cm}\).

2. Calculate the multiplication factor for each dimension by dividing the dimensions of the larger prism by the corresponding dimensions of the smaller prism:

- Factor for the length:
[tex]\[ \text{Factor}_{\text{length}} = \frac{14.7 \text{ cm}}{4.2 \text{ cm}} = 3.5 \][/tex]

- Factor for the width:
[tex]\[ \text{Factor}_{\text{width}} = \frac{20.3 \text{ cm}}{5.8 \text{ cm}} = 3.5 \][/tex]

- Factor for the height:
[tex]\[ \text{Factor}_{\text{height}} = \frac{33.6 \text{ cm}}{9.6 \text{ cm}} = 3.5 \][/tex]

3. Verify that the factors are consistent across all dimensions:
Each multiplication factor calculated (length, width, height) results in a factor of approximately \(3.5\).

4. Conclude the common factor:
Since the common factor for multiplying each dimension of the smaller prism to get the corresponding dimension of the larger prism is consistent and equal to \(3.5\), the answer is:

[tex]\[ 3.5 = 3 \frac{1}{2} \][/tex]

Thus, each dimension of the smaller prism is multiplied by [tex]\(3 \frac{1}{2}\)[/tex] to produce the corresponding dimensions of the larger prism.

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