Question

The areas of three adjacent faces of a rectangular box which meet in a point are known. The product of these areas is equal to $$$$
A. the volume of the box $$$$
B. twice the volume of the box $$$$
C. the square of the volume of the box $$$$
D. the cube root of the volume of the box $$$$

Answer 1

Hint: We denote the denote the vertices of the cuboid shaped rectangular box as A, B, C, D, E, F, G, H and three different sides length, breadth and height denoted $l,$$b$ and $h$. We find surface area of any three adjacent surfaces using the formula for area of a rectangle and find their product. We compare the product with volume of the cuboid box $V=lbh$ to choose the correct option. $$$$

Complete step by step answer:
We know that a cuboid is a three dimensional object with six rectangular faces joined by 8 vertices. It has three different types of sides called length, breadth and height denoted $l,$$b$ and $h$. $$$$
The amount of space contained by a three dimensional object is measured by the quantity called volume. The amount of space that is occupied by a cuboid is the product of length, breadth and height. Mathematically, volume denoted as $V$of a cuboid is
$$V=l\times b\times h=lbh....\left(1\right)$$

 Let us denote the vertices of the cuboid as A, B, C, D, E, F, G, H. We are going to call two rectangular surfaces adjacent when they share a common vertex. We have the rectangular surfaces ABCD, DCFG and ADGH share the common vertex D. Let us assign
$$\begin{array}{rl}& AB=HE=GF=CD=l\\ & AD=BC=EF=GH=b\\ & AH=GD=BE=CF=h\end{array}$$
We are given the question that the areas of three adjacent faces of a rectangular box which meet in a point are known. The rectangular face is the product of its different sides. So the areas of three adjacent faces are
$$\begin{array}{rl}& \text{Area of }ABCD=AB\times BC=l\times b=lb\\ & \text{Area of }DCFG=DC\times GD=h\times l=hl\\ & \text{Area of }ADGH=GH\times GD=l\times b=lb\end{array}$$
So the product surface areas of the three adjacent faces is
$$\begin{array}{rl}& \text{Area of }ABCD\times \text{Area of }ABCD\times \text{Area of }ABCD\\ & =lb\times bh\times hl=l^2{b}^2{h}^2={\left(lbh\right)}^2\end{array}$$
.We use value for equation (1) and have;
$$\text{Area of }ABCD\times \text{Area of }ABCD\times \text{Area of }ABCD={\left(lbh\right)}^2=V^2$$
The product of these areas is equal to the square of the volume.

So, the correct answer is “Option C”.

Note: A cube is cuboid with all sides of equal length which means $l=b=h=a$ and unlike the areas of the faces of a cuboid , the areas of faces of the cube are equal. The total surface area of the rectangular box will be $2(lb+bh+hl)$and length of the space diagonal will be $\sqrt{l^2+b^2+h^2}$.The total surface of cube is $6a^2$ and the volume is $a^3$.