Jeremy will fill a rectangular crate that has inside dimensions of 18 inches by 15 inches by 9 inches with cubical tiles, each with edge lengths of 3 inches. If the tiles are packaged in sets of 8, how many packages will jeremy needs to completely fill the crate?

Options

A. `11`
B. `12`
C. `90`
D. `101`

Answer

To solve the problem step by step, we need to calculate the volume of the rectangular crate, the volume of one cubical tile, and then determine how many packages of tiles Jeremy will need to fill the crate completely.

### Step 1: Calculate the Volume of the Rectangular Crate
The volume \( V \) of a rectangular crate can be calculated using the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given the dimensions of the crate are 18 inches, 15 inches, and 9 inches, we can substitute these values into the formula:
\[ V = 18 \, \text{inches} \times 15 \, \text{inches} \times 9 \, \text{inches} \]
Calculating this:
\[ V = 18 \times 15 = 270 \, \text{square inches} \]
\[ V = 270 \times 9 = 2430 \, \text{cubic inches} \]

### Step 2: Calculate the Volume of One Cubical Tile
The volume \( V_t \) of a cube is calculated using the formula:
\[ V_t = \text{edge length}^3 \]
Given that the edge length of the cubical tile is 3 inches:
\[ V_t = 3 \, \text{inches} \times 3 \, \text{inches} \times 3 \, \text{inches} = 3^3 = 27 \, \text{cubic inches} \]

### Step 3: Calculate the Total Number of Cubical Tiles Needed
To find the total number of tiles \( N \) needed to fill the crate, we divide the volume of the crate by the volume of one tile:
\[ N = \frac{V}{V_t} = \frac{2430 \, \text{cubic inches}}{27 \, \text{cubic inches}} \]
Calculating this:
\[ N = 90 \]
So, Jeremy needs 90 tiles to fill the crate.

### Step 4: Calculate the Number of Packages Required
Since the tiles are packaged in sets of 8, we need to determine how many packages \( P \) are required:
\[ P = \frac{N}{8} = \frac{90}{8} = 11.25 \]
Since Jeremy cannot buy a fraction of a package, he will need to round up to the nearest whole number:
\[ P = 12 \]

### Final Answer
Jeremy needs **12 packages** of tiles to completely fill the crate.

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