The areas of the faces of a rectangular box are 48 m2, 96 m2, and 288 m2.

A second box is cubical and each of its faces has area 16 m2.
Find the ratio of the volume of the first box to the volume of the second box.

let the sides of the first box be

a, b , and c m
so
ab = 48 #1
ac = 96 #2
bc = 288 #3

divide #1 by #2 --> b/c = 1/2 or c = 2b
divide #2 by #3 --> a/b = 1/3 or b = 3a
divide #1 by #3 --> a/c = 1/6 or c = 6a

we know ab = 48
a(3a) = 48
3a^2 = 48
a^2 = 16
a = 4
then b = 12
and c = 24

the volume of the first box must be
4x12x24 or 1152 m^3

the second box must have each side as 4 m
so its volume is 4x4x4 = 64 m^3

so the ratio of theri volumes = 1152:64
= 18:1

To find the ratio of the volume of the first box to the volume of the second box, we first need to determine the dimensions of each box.

Let's start with the first box. Since it is a rectangular box, it has three different face areas: 48 m^2, 96 m^2, and 288 m^2. Let's label the dimensions of the box as length (L), width (W), and height (H).

We can set up the following equations using the given areas:

2(LW) + 2(LH) + 2(WH) = 48
2(LW) + 2(LH) + 2(WH) = 96
2(LW) + 2(LH) + 2(WH) = 288

Simplifying each equation:
LW + LH + WH = 24
LW + LH + WH = 48
LW + LH + WH = 144

Now, a second box is cubical, meaning all sides are equal. Let's label the length of each side as S.

The area of one face of the second box is given as 16 m^2. Since all faces are equal, we have:

6S^2 = 16

Simplifying this equation:
S^2 = 16/6
S^2 = 8/3

To find the volume of each box, we multiply the dimensions together. So, for the first box:

Volume of first box = LWH

For the second box:

Volume of second box = S^3

Now, since we don't know the exact values of L, W, H, and S, we can't find the exact ratio of the volumes. However, we can find the ratio of the volumes in terms of the sides.

Ratio of volumes = (LWH) / (S^3)

Therefore, in order to find the desired ratio, we need to determine the values of L, W, H, and S and then plug them into the equation.