Answer:
2.65 inches
Step-by-step explanation:
The length of the toothpick is 3 inches.
The of the base of the box is 1 inch x 1 inch.
Let the height of the box is h inch, so, the dimension of the box is
1 inch x 1 inch x h inch.
The toothpick must be placed diagonally for the minimum height of the box, as shown in the figure,
So, the length of the longest diagonal of the box
[tex]d=\sqrt {1^2+1^2+h^2}[/tex]
As the length of the toothpick is 3 inches, so d= 3
[tex]\Rightarrow 3=\sqrt {2+h^2}[/tex]
[tex]\Rightarrow 9=2+h^2[/tex]
[tex]\Rightarrow h^2=9-2=7[/tex]
[tex]\Rightarrow h = \sqrt 7=2.65[/tex] inches.
Hence, the shortest possible length of the box is 2.65 inches.
