When length of each side of a cube is increased by 3cm, its volume is increased by `2457cm^(3)`. Find its side. How much will its volume decrease, if length of each side of it is reduced by 20%?

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We need to find the original side length of a cube when its volume increases by 2457 cm³ after increasing each side by 3 cm. Then, we will calculate how much the volume decreases if the side length is reduced by 20%.

### Step 2: Set up the equation
Let the original side length of the cube be \( a \) cm. The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]

When the side length is increased by 3 cm, the new side length becomes \( a + 3 \) cm. The new volume is:
\[ V' = (a + 3)^3 \]

According to the problem, the increase in volume is 2457 cm³:
\[ (a + 3)^3 - a^3 = 2457 \]

### Step 3: Expand the equation
Using the binomial expansion:
\[ (a + 3)^3 = a^3 + 3 \cdot a^2 \cdot 3 + 3 \cdot a \cdot 3^2 + 3^3 \]
This simplifies to:
\[ (a + 3)^3 = a^3 + 9a^2 + 27a + 27 \]

Now, substituting this back into our equation:
\[ a^3 + 9a^2 + 27a + 27 - a^3 = 2457 \]
This simplifies to:
\[ 9a^2 + 27a + 27 = 2457 \]

### Step 4: Rearrange the equation
Now, we can rearrange this equation:
\[ 9a^2 + 27a + 27 - 2457 = 0 \]
\[ 9a^2 + 27a - 2430 = 0 \]

### Step 5: Simplify the equation
We can divide the entire equation by 9 to simplify:
\[ a^2 + 3a - 270 = 0 \]

### Step 6: Factor the quadratic equation
To factor the quadratic equation \( a^2 + 3a - 270 = 0 \), we look for two numbers that multiply to \(-270\) and add to \(3\). These numbers are \(18\) and \(-15\):
\[ (a + 18)(a - 15) = 0 \]

### Step 7: Solve for \( a \)
Setting each factor to zero gives us:
1. \( a + 18 = 0 \) → \( a = -18 \) (not valid since side length cannot be negative)
2. \( a - 15 = 0 \) → \( a = 15 \)

Thus, the original side length of the cube is \( a = 15 \) cm.

### Step 8: Calculate the original volume
Now, we can calculate the volume of the cube with side length \( 15 \) cm:
\[ V = a^3 = 15^3 = 3375 \text{ cm}^3 \]

### Step 9: Calculate the new side length after a 20% reduction
To find the new side length after reducing by 20%, we calculate:
\[ \text{New side length} = a - 0.2a = 0.8a = 0.8 \times 15 = 12 \text{ cm} \]

### Step 10: Calculate the new volume
Now, we calculate the volume of the cube with the new side length:
\[ V' = (12)^3 = 1728 \text{ cm}^3 \]

### Step 11: Calculate the decrease in volume
Finally, we find the decrease in volume:
\[ \text{Decrease in volume} = V - V' = 3375 - 1728 = 1647 \text{ cm}^3 \]

### Final Answer
The original side length of the cube is \( 15 \) cm, and the volume decreases by \( 1647 \) cm³ when the side length is reduced by 20%.

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