Marcus's yard has a large square sandbox serving as a cactus bed. The side lenghts of the box are 24 feet, and the box is currently one-third full of sand from the bottom up. Mearcus purchases 480 cubic feet of sand, which, when added to the box, will completely fill it. How many inches deep is the cactus bed ?

Options

A. 10
B. 15
C. 20
D. 25

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the volume of the sandbox
The sandbox is a square box with side lengths of 24 feet. The volume \( V \) of a box is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Since it is a square box, the length and width are both 24 feet. Let \( H \) be the height of the box. Therefore, the volume can be expressed as:
\[ V = 24 \times 24 \times H \]

### Step 2: Determine the current volume of sand in the box
The box is currently one-third full of sand. Therefore, the volume of sand currently in the box is:
\[ \text{Current Volume of Sand} = \frac{1}{3} V = \frac{1}{3} (24 \times 24 \times H) = \frac{576H}{3} = 192H \text{ cubic feet} \]

### Step 3: Determine the empty volume in the box
Since the box is one-third full, the volume that is empty is:
\[ \text{Empty Volume} = V - \text{Current Volume of Sand} = V - 192H \]
Substituting the volume \( V \):
\[ \text{Empty Volume} = (24 \times 24 \times H) - 192H = 576H - 192H = 384H \text{ cubic feet} \]

### Step 4: Relate the empty volume to the sand purchased
Marcus purchases 480 cubic feet of sand to fill the box completely. Therefore, we can set up the equation:
\[ \text{Empty Volume} = 480 \text{ cubic feet} \]
Thus, we have:
\[ 384H = 480 \]

### Step 5: Solve for the height \( H \)
To find \( H \), we divide both sides by 384:
\[ H = \frac{480}{384} = \frac{5}{4} = 1.25 \text{ feet} \]

### Step 6: Convert the height from feet to inches
Since the question asks for the depth in inches, we convert feet to inches:
\[ H = 1.25 \text{ feet} \times 12 \text{ inches/foot} = 15 \text{ inches} \]

### Final Answer
The depth of the cactus bed is **15 inches**.

---