If the edge of a cube is increased by 4 cm , the volume will increase by `988 cm^(3)` . Then the original length of each edge of the cube is

Options

A. 7 cm
B. 9 cm
C. 6 cm
D. 8 cm

Answer

To solve the problem step by step, let's denote the original length of each edge of the cube as \( a \) cm.

### Step 1: Write the formula for the volume of a cube
The volume \( V \) of a cube with edge length \( a \) is given by:
\[ V = a^3 \]

### Step 2: Write the volume of the cube after increasing the edge length
If the edge length is increased by 4 cm, the new edge length becomes \( a + 4 \) cm. The volume of the new cube is:
\[ V' = (a + 4)^3 \]

### Step 3: Set up the equation for the increase in volume
According to the problem, the increase in volume is given as \( 988 \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ V' - V = 988 \]
Substituting the volumes we found:
\[ (a + 4)^3 - a^3 = 988 \]

### Step 4: Expand the left-hand side
Now, we expand \( (a + 4)^3 \):
\[ (a + 4)^3 = a^3 + 3a^2(4) + 3a(4^2) + 4^3 \]
This simplifies to:
\[ a^3 + 12a^2 + 48a + 64 \]
So, the equation becomes:
\[ (a^3 + 12a^2 + 48a + 64) - a^3 = 988 \]
This simplifies to:
\[ 12a^2 + 48a + 64 = 988 \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation:
\[ 12a^2 + 48a + 64 - 988 = 0 \]
This simplifies to:
\[ 12a^2 + 48a - 924 = 0 \]

### Step 6: Simplify the equation
We can divide the entire equation by 12 to simplify it:
\[ a^2 + 4a - 77 = 0 \]

### Step 7: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ (a + 11)(a - 7) = 0 \]

### Step 8: Solve for \( a \)
Setting each factor to zero gives us:
1. \( a + 11 = 0 \) → \( a = -11 \) (not valid since edge length cannot be negative)
2. \( a - 7 = 0 \) → \( a = 7 \)

Thus, the original length of each edge of the cube is:
\[ \boxed{7 \, \text{cm}} \]