On each face of a cuboid, the sum of its perimeter and its area is written. Among the six numbers so written, there are three distinct numbers and they are 16, 24 and 31. The volume of the cuboid lies between

Options

A. 7 and 14
B. 14 and 21
C. 21 and 28
D. 28 and 35

Answer

To solve the problem step by step, we will analyze the given information about the cuboid and derive the equations needed to find the volume.

### Step 1: Understand the Problem
We are given that on each face of a cuboid, the sum of its perimeter and area is written. The distinct sums are 16, 24, and 31. We need to find the volume of the cuboid.

### Step 2: Define Variables
Let the dimensions of the cuboid be:
- Length = L
- Breadth = B
- Height = H

### Step 3: Write the Expressions for Each Face
The perimeter and area for each face of the cuboid can be expressed as follows:
1. Face with dimensions L and H:
- Perimeter = 2(L + H)
- Area = L * H
- Sum = 2(L + H) + L * H

2. Face with dimensions L and B:
- Perimeter = 2(L + B)
- Area = L * B
- Sum = 2(L + B) + L * B

3. Face with dimensions B and H:
- Perimeter = 2(B + H)
- Area = B * H
- Sum = 2(B + H) + B * H

### Step 4: Set Up the Equations
From the problem, we have three distinct sums:
1. \( 2(L + H) + LH = 16 \)
2. \( 2(L + B) + LB = 24 \)
3. \( 2(B + H) + BH = 31 \)

### Step 5: Rearranging the Equations
We can rearrange each equation to express them in terms of one variable:
1. \( LH + 2L + 2H = 16 \)
2. \( LB + 2L + 2B = 24 \)
3. \( BH + 2B + 2H = 31 \)

### Step 6: Solve for One Variable
Let’s solve for \( B \) from the third equation:
\[ BH + 2B + 2H = 31 \implies B(H + 2) = 31 - 2H \implies B = \frac{31 - 2H}{H + 2} \]

### Step 7: Substitute B in the Second Equation
Substituting \( B \) into the second equation:
\[ L \left( \frac{31 - 2H}{H + 2} \right) + 2L + 2 \left( \frac{31 - 2H}{H + 2} \right) = 24 \]

### Step 8: Solve for L
This will give us a complicated equation in terms of \( L \) and \( H \). We can also express \( L \) in terms of \( H \) using the first equation:
\[ L = \frac{16 - 2H}{H + 2} \]

### Step 9: Substitute L in the Equation
Substituting \( L \) into the equation derived from the second equation will yield a polynomial in \( H \).

### Step 10: Solve the Polynomial
After simplification, we will arrive at a quadratic equation in \( H \). Solving this quadratic equation will give us the possible values for \( H \).

### Step 11: Find Values for B and L
Once we have \( H \), we can substitute back to find \( B \) and \( L \).

### Step 12: Calculate the Volume
The volume \( V \) of the cuboid is given by:
\[ V = L \times B \times H \]

### Step 13: Determine the Range of Volume
Finally, we will check the calculated volume against the options given in the problem to determine which range it falls into.