Question

convert 454 in³ into cm³ and into ml. (2.54 cm = 1 in). put in the units for the conversion factor. (454 in³)(----------) = cm³ options for the fractions numerator and denominator: in, cm, in³, cm³ (repeated for the conversion setup)

Explanation:

Step1: Understand the conversion factor for volume

Since \( 1 \text{ in} = 2.54 \text{ cm} \), for volume, we cube both sides. So \( (1 \text{ in})^3=(2.54 \text{ cm})^3 \), which means \( 1 \text{ in}^3=(2.54)^3 \text{ cm}^3 \approx 16.387 \text{ cm}^3 \). The conversion factor to convert cubic inches to cubic centimeters is \( \frac{(2.54 \text{ cm})^3}{1 \text{ in}^3}=\frac{16.387 \text{ cm}^3}{1 \text{ in}^3} \). So the correct conversion factor has units \( \frac{\text{cm}^3}{\text{in}^3} \) (or when considering the cube of the length conversion, the numerator is \( \text{cm}^3 \) and denominator is \( \text{in}^3 \)). Looking at the options, the third option (with numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \) - wait, no, let's re - examine the options. Wait, the options are structured as numerator unit over denominator unit. Wait, the problem is about putting the units for the conversion factor. Let's think again. The conversion from \( \text{in}^3 \) to \( \text{cm}^3 \) is based on \( (1 \text{ in}) = 2.54 \text{ cm} \), so \( 1 \text{ in}^3=(2.54 \text{ cm})^3 \). So to convert \( 454 \text{ in}^3 \) to \( \text{cm}^3 \), we use the conversion factor \( \frac{(2.54 \text{ cm})^3}{1 \text{ in}^3} \), which has units \( \frac{\text{cm}^3}{\text{in}^3} \). So among the options, the one with numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \) (the third option in the list, where the numerator is \( \text{cm}^3 \) and denominator is \( \text{in}^3 \)? Wait, looking at the options:

First option: numerator in, denominator? Wait, no, the options are:

First: \( \frac{\text{in}}{\text{cm}} \)? No, the first option's numerator is in, denominator? Wait, the user's options are:

1. \( \frac{\text{in}}{\text{cm}} \) (wait, no, the first option is written as numerator in, denominator? Wait, the problem shows:

First option:
\( (454 \text{ in}^3)(\frac{\text{in}}{\text{...}}) \)? No, the first option's numerator is in, denominator is cm? Wait, no, the user's diagram:

First option:
\( (454 \text{ in}^3)(\frac{\text{in}}{\text{cm}}) \)? No, the first option's numerator is in, denominator is cm? Wait, no, the user's options are:

Option 1: numerator in, denominator (maybe cm? The text is a bit unclear, but let's think about unit conversion. To convert \( \text{in}^3 \) to \( \text{cm}^3 \), we need to cube the length conversion. Since \( 1 \text{ in} = 2.54 \text{ cm} \), then \( 1 \text{ in}^3=(2.54 \text{ cm})^3 \). So the conversion factor is \( \frac{(2.54 \text{ cm})^3}{1 \text{ in}^3} \), which has units \( \frac{\text{cm}^3}{\text{in}^3} \). So among the options, the third option (the one with numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \)) is correct. Wait, the third option in the user's list:

Third option:
\( (454 \text{ in}^3)(\frac{\text{cm}^3}{\text{in}^3}) \)? Wait, no, the third option's numerator is \( \text{cm}^3 \), denominator is \( \text{in}^3 \)? Wait, the user's options:

Looking at the user's problem:

First option: numerator in, denominator (maybe cm? The first option is:

\( (454 \text{ in}^3)(\frac{\text{in}}{\text{cm}}) \)? No, the first option's numerator is in, denominator is cm? That would be wrong.

Second option: numerator \( \text{in}^3 \), denominator \( \text{cm} \)? No, that's wrong.

Third option: numerator \( \text{cm}^3 \), denominator \( \text{in}^3 \)? Wait, no, the third option's numerator is \( \text{cm}^3 \), denominator is \( \text{in}^3 \)? Wait, the third option is:

\( (454 \text{ in}^3)(\frac{\text{cm}^3}{\text{in}^3}) \)? Wait, no, the third option's numerator is \( \text{cm}^3 \), denominator is \( \text{in}^3 \)? Wait, the user's problem:

Third option:

\( (454 \text{ in}^3)(\frac{\text{cm}^3}{\text{in}^3}) \)? No, the third option's numerator is \( \text{cm}^3 \), denominator is \( \text{in}^3 \)? Wait, the correct conversion factor for volume from \( \text{in}^3 \) to \( \text{cm}^3 \) is \( \frac{(2.54 \text{ cm})^3}{1 \text{ in}^3}=\frac{\text{cm}^3}{\text{in}^3} \) (since \( (2.54 \text{ cm})^3 \) is \( \text{cm}^3 \) and the denominator is \( \text{in}^3 \)). So the correct option is the one with numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \), which is the third option (the one where the numerator is \( \text{cm}^3 \) and denominator is \( \text{in}^3 \)). Wait, the user's options:

First option: numerator in, denominator cm?

Second option: numerator \( \text{in}^3 \), denominator cm?

Third option: numerator \( \text{cm}^3 \), denominator \( \text{in}^3 \)?

Fourth option: numerator in, denominator \( \text{cm}^3 \)?

No, let's re - express. The conversion factor for length is \( \frac{2.54 \text{ cm}}{1 \text{ in}} \). For volume, we cube both sides: \( (\frac{2.54 \text{ cm}}{1 \text{ in}})^3=\frac{(2.54)^3 \text{ cm}^3}{1 \text{ in}^3} \). So the conversion factor for volume has units \( \frac{\text{cm}^3}{\text{in}^3} \). So when we multiply \( 454 \text{ in}^3 \) by this conversion factor, the \( \text{in}^3 \) units cancel out, leaving \( \text{cm}^3 \). So the correct conversion factor's units are \( \frac{\text{cm}^3}{\text{in}^3} \), which corresponds to the option where the numerator is \( \text{cm}^3 \) and the denominator is \( \text{in}^3 \) (the third option in the list, as per the user's diagram: the third option has numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \)? Wait, the user's third option is:

\( (454 \text{ in}^3)(\frac{\text{cm}^3}{\text{in}^3}) \)? No, the third option's numerator is \( \text{cm}^3 \), denominator is \( \text{in}^3 \). So the correct option is the third one (the one with numerator \( \text{cm}^3 \) and denominator \( \text{in}^3 \)).

Step2: Confirm the unit conversion logic

To convert cubic inches to cubic centimeters, we use the cube of the length conversion factor. Since \( 1 \text{ in} = 2.54 \text{ cm} \), cubing both sides gives \( 1 \text{ in}^3=(2.54)^3 \text{ cm}^3 \). The conversion factor is \( \frac{(2.54 \text{ cm})^3}{1 \text{ in}^3} \), which has units of \( \frac{\text{cm}^3}{\text{in}^3} \). So when we multiply \( 454 \text{ in}^3 \) by this conversion factor, the \( \text{in}^3 \) units cancel, and we get \( \text{cm}^3 \).

Answer:

The correct option is the one with the conversion factor having units \( \frac{\text{cm}^3}{\text{in}^3} \) (the third option in the provided list, where the numerator is \( \text{cm}^3 \) and the denominator is \( \text{in}^3 \)). To calculate the numerical value:

First, calculate \( (2.54)^3 = 2.54\times2.54\times2.54 = 16.387064 \).

Then, \( 454 \text{ in}^3\times\frac{16.387064 \text{ cm}^3}{1 \text{ in}^3}=454\times16.387064 \text{ cm}^3\approx7440 \text{ cm}^3 \) (and since \( 1 \text{ cm}^3 = 1 \text{ mL} \), it is also \( 7440 \text{ mL} \)). But for the unit conversion factor's units, the correct option is the one with \( \frac{\text{cm}^3}{\text{in}^3} \) (the third option in the given choices).