Question
What's the longest pen that coulod fit if the height of the box is 8 inch and diagonal of the base of the box is 6 inch
Answers
GPT-4o mini
To determine the longest pen that could fit inside a box with a height of 8 inches and a diagonal of the base of 6 inches, we can use the Pythagorean theorem.
1. First, let’s denote the height of the box as \(h = 8\) inches.
2. The diagonal of the base of the box is given as \(d = 6\) inches.
The diagonal of the base (\(d\)) can be expressed in terms of the length (\(l\)) and width (\(w\)) of the box using the Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2}
\]
Given that \(d = 6\):
\[
l^2 + w^2 = 6^2 = 36
\]
To find the longest pen that can fit in the box, we need to find the space diagonal (\(D\)) of the box, which can be calculated using the Pythagorean theorem in three dimensions:
\[
D = \sqrt{l^2 + w^2 + h^2}
\]
We know that:
\[
l^2 + w^2 = 36
\]
and \(h = 8\):
\[
D = \sqrt{36 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
Thus, the longest pen that could fit inside the box measures **10 inches**.
1. First, let’s denote the height of the box as \(h = 8\) inches.
2. The diagonal of the base of the box is given as \(d = 6\) inches.
The diagonal of the base (\(d\)) can be expressed in terms of the length (\(l\)) and width (\(w\)) of the box using the Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2}
\]
Given that \(d = 6\):
\[
l^2 + w^2 = 6^2 = 36
\]
To find the longest pen that can fit in the box, we need to find the space diagonal (\(D\)) of the box, which can be calculated using the Pythagorean theorem in three dimensions:
\[
D = \sqrt{l^2 + w^2 + h^2}
\]
We know that:
\[
l^2 + w^2 = 36
\]
and \(h = 8\):
\[
D = \sqrt{36 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
Thus, the longest pen that could fit inside the box measures **10 inches**.