Question

Use complete sentences to explain the process you would use to find the volume of the shipping box.(Trying to help my son with this)

Answer 1

Part A)

The given shipping box is a cuboid.

Recall that the longest length of the cuboid is diagonal.

The length of the longest item that fits inside the shipping box is the measure of the diagonal of the given box.

Given that measure breadth=16 inches and measure height = 12 inches.

Recall the formula for the diagonal d of the cuboid is

`d=\sqrt[]{l^2+b^2+h^2}`

We need to find the measure of the length of the cuboid.

Consider the base of the cuboid which is in rectangle shape.

Here breadth of the rectangle is 16 inches and diagonal of the rectangle is 24 inches.

Recall the formula for the diagonal of the rectangle is

`diagonal_{}=\sqrt[]{l^2+b^2}`

Substitute diagonal =24 inches and breath =16 inches, we get

`24_{}=\sqrt[]{l^2+16^2}`

`24_{}=\sqrt[]{l^2+256}`

Taking square on both sides, we get

`24^2_{}=l^2+256`

`576-256=l^2`

`320=l^2`

Taking square root on both sides, we get

`\sqrt[]{320}=l`
`l=17.89\text{ inches}`

Now, substitute l=17.89, b=16, and h=12 in the diagonal of the cuboid equation to find the diagonal of the cuboid.

`d^{}=\sqrt[]{17.89^2+16^2+12^2}`

`d^{}=\sqrt[]{320+256+144}=\sqrt[]{720}=26.83\text{ inches}`

Hence the length of the longest item that fits inside the shipping box is 26.8 inches.

Part B)

Consider the length l=17.89 inches, b=16 inches, and height h=12 inches.

Recall the formula for the volume of the cuboid is

`V=l* w* h`

Substitute the length l=17.89 inches, b=16 inches, and height h=12 inches, we get

`V=17.89*16*12`
`V=3434.88inches^3`

Hence the volume of the given shipping box is 3434.88 cubic inches.