A rectangular box is to be made form a sheet of 24 inch length and 9 inch width cutting out identical squares of side length x from the four corners and turning up the sides
What is the maximum volume of the box ?

Options

A. 200 cubic inch
B. 400 cubic inch
C. 100 cubic inch
D. none of these

Explanation

To find the maximum volume of the rectangular box made from a sheet of dimensions 24 inches by 9 inches by cutting out squares of side length \( x \) from each corner, we can follow these steps:

### Step 1: Define the dimensions of the box
When we cut out squares of side length \( x \) from each corner, the dimensions of the box will change as follows:
- **Length**: \( L = 24 - 2x \)
- **Width**: \( W = 9 - 2x \)
- **Height**: \( H = x \)

### Step 2: Write the volume function
The volume \( V \) of the box can be expressed as:
\[ V = L \times W \times H = (24 - 2x)(9 - 2x)(x) \]
Expanding this, we get:
\[ V = (24 - 2x)(9 - 2x)x \]

### Step 3: Expand the volume expression
First, expand \( (24 - 2x)(9 - 2x) \):
\[ (24 - 2x)(9 - 2x) = 216 - 48x + 4x^2 \]
Now, multiply by \( x \):
\[ V = x(216 - 48x + 4x^2) = 4x^3 - 48x^2 + 216x \]

### Step 4: Differentiate the volume function
To find the maximum volume, we need to differentiate \( V \) with respect to \( x \):
\[ \frac{dV}{dx} = 12x^2 - 96x + 216 \]

### Step 5: Set the derivative to zero
Set the derivative equal to zero to find critical points:
\[ 12x^2 - 96x + 216 = 0 \]
Dividing the entire equation by 12:
\[ x^2 - 8x + 18 = 0 \]

### Step 6: Factor the quadratic equation
Now, we can factor the quadratic:
\[ (x - 9)(x - 2) = 0 \]
Thus, the solutions are:
\[ x = 9 \quad \text{and} \quad x = 2 \]

### Step 7: Determine the nature of critical points
To determine which critical point gives the maximum volume, we can use the second derivative test:
\[ \frac{d^2V}{dx^2} = 24x - 96 \]
Evaluate at \( x = 2 \):
\[ \frac{d^2V}{dx^2} \bigg|_{x=2} = 24(2) - 96 = -48 \quad (\text{negative, hence maximum}) \]
Evaluate at \( x = 9 \):
\[ \frac{d^2V}{dx^2} \bigg|_{x=9} = 24(9) - 96 = 96 \quad (\text{positive, hence minimum}) \]

### Step 8: Calculate the maximum volume
Now, substitute \( x = 2 \) back into the volume function:
\[ V = (24 - 2(2))(9 - 2(2))(2) = (24 - 4)(9 - 4)(2) = (20)(5)(2) = 200 \text{ cubic inches} \]

### Final Answer
The maximum volume of the box is \( 200 \) cubic inches.

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