Question
If you want the diagonal, d, to be at least 41.5 inches, what would be a possible length, width, and height for the box?
(1 point)
Responses
23 inches, 24 inches, 25 inches
23 inches, 24 inches, 25 inches
21 inches, 24 inches, 26 inches
21 inches, 24 inches, 26 inches
20 inches, 20 inches, 27 inches
20 inches, 20 inches, 27 inches
20 inches, 21 inches, 26 inches
Answers
GPT-4o mini
To determine if any of these sets of dimensions for a box will result in a diagonal of at least 41.5 inches, we can use the formula for the diagonal \( d \) of a rectangular box:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
We will calculate the diagonal for each option:
1. **Dimensions: 23 inches, 24 inches, 25 inches**
\[
d = \sqrt{23^2 + 24^2 + 25^2} = \sqrt{529 + 576 + 625} = \sqrt{1730} \approx 41.6 \text{ inches}
\]
(This meets the requirement.)
2. **Dimensions: 21 inches, 24 inches, 26 inches**
\[
d = \sqrt{21^2 + 24^2 + 26^2} = \sqrt{441 + 576 + 676} = \sqrt{1693} \approx 41.1 \text{ inches}
\]
(This does not meet the requirement.)
3. **Dimensions: 20 inches, 20 inches, 27 inches**
\[
d = \sqrt{20^2 + 20^2 + 27^2} = \sqrt{400 + 400 + 729} = \sqrt{1529} \approx 39.1 \text{ inches}
\]
(This does not meet the requirement.)
4. **Dimensions: 20 inches, 21 inches, 26 inches**
\[
d = \sqrt{20^2 + 21^2 + 26^2} = \sqrt{400 + 441 + 676} = \sqrt{1517} \approx 38.9 \text{ inches}
\]
(This does not meet the requirement.)
From these calculations, the only set of dimensions that results in a diagonal of at least 41.5 inches is:
**23 inches, 24 inches, 25 inches.**
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
We will calculate the diagonal for each option:
1. **Dimensions: 23 inches, 24 inches, 25 inches**
\[
d = \sqrt{23^2 + 24^2 + 25^2} = \sqrt{529 + 576 + 625} = \sqrt{1730} \approx 41.6 \text{ inches}
\]
(This meets the requirement.)
2. **Dimensions: 21 inches, 24 inches, 26 inches**
\[
d = \sqrt{21^2 + 24^2 + 26^2} = \sqrt{441 + 576 + 676} = \sqrt{1693} \approx 41.1 \text{ inches}
\]
(This does not meet the requirement.)
3. **Dimensions: 20 inches, 20 inches, 27 inches**
\[
d = \sqrt{20^2 + 20^2 + 27^2} = \sqrt{400 + 400 + 729} = \sqrt{1529} \approx 39.1 \text{ inches}
\]
(This does not meet the requirement.)
4. **Dimensions: 20 inches, 21 inches, 26 inches**
\[
d = \sqrt{20^2 + 21^2 + 26^2} = \sqrt{400 + 441 + 676} = \sqrt{1517} \approx 38.9 \text{ inches}
\]
(This does not meet the requirement.)
From these calculations, the only set of dimensions that results in a diagonal of at least 41.5 inches is:
**23 inches, 24 inches, 25 inches.**