Brand X paint costs $14 per gallon, and 1 gallon provides coverage of an area of at most 150 square feet. What is the minimum cost of the amount of brand X paint needed to cover the four walls of a rectangular room that is 12 feet wide, 16 feet long, and 8 feet high?

Answer

To find the minimum cost of the amount of Brand X paint needed to cover the four walls of a rectangular room, we will follow these steps:

### Step 1: Calculate the lateral surface area of the room.
The formula for the lateral surface area (A) of a rectangular room (cuboid) is given by:
\[ A = 2 \times h \times (l + b) \]
where:
- \( h \) = height of the room
- \( l \) = length of the room
- \( b \) = width of the room

Given:
- \( h = 8 \) feet
- \( l = 16 \) feet
- \( b = 12 \) feet

Substituting the values:
\[ A = 2 \times 8 \times (16 + 12) \]
\[ A = 2 \times 8 \times 28 \]
\[ A = 16 \times 28 \]
\[ A = 448 \text{ square feet} \]

### Step 2: Determine how many gallons of paint are needed.
Each gallon of paint covers a maximum area of 150 square feet. To find out how many gallons are required, we divide the total area by the coverage per gallon:
\[ \text{Gallons required} = \frac{A}{\text{Coverage per gallon}} \]
\[ \text{Gallons required} = \frac{448}{150} \]
\[ \text{Gallons required} = 2.9867 \]

Since we cannot purchase a fraction of a gallon, we round up to the nearest whole number:
\[ \text{Gallons required} = 3 \text{ gallons} \]

### Step 3: Calculate the total cost of the paint.
The cost of one gallon of paint is $14. Therefore, the total cost for 3 gallons is:
\[ \text{Total cost} = \text{Gallons required} \times \text{Cost per gallon} \]
\[ \text{Total cost} = 3 \times 14 \]
\[ \text{Total cost} = 42 \text{ dollars} \]

### Final Answer:
The minimum cost of the amount of Brand X paint needed to cover the four walls of the room is **$42**.

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