Question

Calculate the length of the longest rod that can fit in a box measuring 100 centimeters by 130 centimeters by 400 centimeters. Round to the nearest tenth.

Answer 1

The diagonal measures about 431.9 cm.

To solve this problem

We can use the formula for the diagonal of a rectangular box:

Diagonal =
`sqrt(length^2 + width^2 + height^2)`

Considering the size:

• Length = 100 centimeters
• Width = 130 centimeters
• Height = 400 centimeters

Substitute these values into the formula:

Diagonal =
`sqrt(100^2 + 130^2 + 400^2)`

Diagonal =
`sqrt(10000 + 16900 + 160000)`

Diagonal =
`sqrt(186900)`

So, To the closest tenth of a centimeter, the diagonal measures about 431.9 cm.

Answer 2

The given box is a cuboid. A labelled diagram is shown below

Looking at the diagram above, the longest side that a rod can fit in is side AB. To find AB, we would find BC first by applying the pythagorean theorem which states that

`\begin{gathered} \text{hypotenuse}^2=oppositeside^2+adjacentside^2 \\ \text{Considering right angle triangle BDC, } \\ \text{hypotenuse = BC} \\ \text{opposite side = DC = 100} \\ \text{Adjacent side = BD = 400} \\ BC^2\text{ = }100^2+400^2\text{ = 170000} \\ BC\text{ = }\sqrt[]{170000}\text{ = 412.32 cm} \\ \end{gathered}`

Since we know BC, we would find AB by considering riight angle triangle ABC

`\begin{gathered} AB\text{ = hypotenuse} \\ BC\text{ = adjacent side = 412.3}1 \\ AC\text{ = opposite side = 130} \\ \text{Thus, } \\ AB^2=412.31^2+130^2\text{ = 186899.5361} \\ AB\text{ = }\sqrt[]{186899.5361} \\ AB\text{ = 432.32} \end{gathered}`

Thus, to the nearest tenth, the longest length that a rod can fit is 432.3 centimeters