Answer:
The volume of the shipping box is 3434.6 in³ (to the nearest tenth).
The length of the longest item that will fit inside the shipping box is 26.8 inches (to the nearest tenth).
Step-by-step explanation:
The shipping box can be modelled as a cuboid.
A cuboid is a three-dimensional geometric shape with six rectangular faces and right angles between adjacent faces.
The volume of a cuboid can be calculated by multiplying its length (L), width (W), and height (H) together.
From the given diagram, the width of the cuboid is 16 inches and its height is 12 inches. Therefore, we need to find the measure of its length in order to calculate its volume.
As all sides of a cuboid have interior angles of 90°, and we have been given the face diagonal of the base (24 inches), we can use Pythagoras Theorem to calculate the length (L).
[tex]\begin{aligned}L^2+16^2&=24^2\\L^2+256&=576\\L^2&=320\\L&=\sqrt{320}\\L&=8\sqrt{5}\; \sf in\end{aligned}[/tex]
Substitute L = 8√5, W = 16 and H = 12 into the formula for the volume of a cuboid to calculate the volume of the shipping box:
[tex]\begin{aligned}\sf Volume&=\sf L \cdot W \cdot H\\&=8\sqrt{5} \cdot 16 \cdot 12\\&=128\sqrt{5} \cdot 12\\&=1526\sqrt{5}\\&=3434.60041...\\&=3434.6\; \sf in^3\end{aligned}[/tex]
Therefore, the volume of the shipping box is 3434.6 in³ to the nearest tenth.
[tex]\hrulefill[/tex]
In a cuboid, there are two types of diagonals: face diagonals and body diagonals.
- Face Diagonals: These diagonals connect opposite corners of a face of the cuboid and lie entirely within that face.
- Body Diagonals: These diagonals connect opposite corners of the cuboid, passing through the interior of the cuboid and extending across multiple faces. Body diagonals are longer than face diagonals.
The body diagonal of a cuboid is the longest line that can be drawn inside the cuboid. Therefore, to find the length of the longest item that will fit inside the shipping box, we need to calculate the body diagonal of the cuboid.
The formula for the body diagonal of a cuboid is:
[tex]\sf Body \;diagonal=\sqrt{L^2+W^2+H^2}[/tex]
Substitute L = 8√5, W = 16 and H = 12 into the formula to find the body diagonal of the cuboid (marked as a red dashed line on the given diagram):
[tex]\begin{aligned}\sf Body \;diagonal&=\sf \sqrt{L^2+W^2+H^2}\\&=\sqrt{(8\sqrt{5})^2+16^2+12^2\\&=\sqrt{320+256+144}\\&=\sqrt{720}\\&=26.8328157...\\&=26.8\; \sf in\;(nearest\;tenth)\end{aligned}[/tex]
Therefore, the length of the longest item that will fit inside the shipping box is 26.8 inches, to the nearest tenth.
