T.J. Is Wallpapering His Bedroom. The Room Has 8 Ft High Ceilings And Is $12 \frac{1}{2}$ Ft Long. If The Perimeter Of The Room Is $48 \frac{1}{2}$ Ft, What Is The Width Of The Room?
Introduction
In this problem, we are given the length and height of a room, as well as its perimeter. We need to use this information to find the width of the room. This is a classic problem in geometry and algebra, and it requires us to use our knowledge of formulas and equations to solve it.
The Problem
T.J. is wallpapering his bedroom, and we need to help him figure out the width of the room. The room has 8 ft high ceilings and is $12 \frac{1}{2}$ ft long. If the perimeter of the room is $48 \frac{1}{2}$ ft, what is the width of the room?
Understanding the Perimeter
The perimeter of a room is the distance around the room. It is calculated by adding up the lengths of all four sides of the room. In this case, the perimeter is given as $48 \frac1}{2}$ ft. We can write this as a mixed number{2} = 48.5$ ft.
Formulating the Equation
Let's denote the width of the room as $w$. Since the room is $12 \frac{1}{2}$ ft long, we can write the length as $12.5$ ft. The perimeter of the room is the sum of the lengths of all four sides, which can be written as:
where $l$ is the length, $w$ is the width, and $P$ is the perimeter.
Substituting the Values
We are given that the length $l = 12.5$ ft and the perimeter $P = 48.5$ ft. We can substitute these values into the equation:
Simplifying the Equation
We can simplify the equation by multiplying the length by 2:
Isolating the Width
We can isolate the width by subtracting 25 from both sides of the equation:
Solving for the Width
We can solve for the width by dividing both sides of the equation by 2:
Calculating the Width
We can calculate the width by dividing 23.5 by 2:
Conclusion
In this problem, we used the formula for the perimeter of a room to find the width of the room. We were given the length and height of the room, as well as its perimeter. We used algebraic manipulations to isolate the width and solve for its value. The width of the room is $11.75$ ft.
Additional Tips and Variations
- If the room has a different shape, such as a rectangle or a square, the formula for the perimeter will be different.
- If the room has a different height, the formula for the perimeter will also be different.
- If the room has a different length, the formula for the perimeter will also be different.
- If the room has a different perimeter, the formula for the width will also be different.
Real-World Applications
This problem has real-world applications in architecture, engineering, and design. When designing a room, it is essential to consider the dimensions of the room, including its length, width, and height. This will help ensure that the room is functional and aesthetically pleasing.
Common Mistakes
- Failing to consider the height of the room when calculating the perimeter.
- Failing to consider the length of the room when calculating the perimeter.
- Failing to use the correct formula for the perimeter of a room.
- Failing to isolate the width correctly.
Conclusion
Q: What is the formula for the perimeter of a room?
A: The formula for the perimeter of a room is:
where $P$ is the perimeter, $l$ is the length, and $w$ is the width.
Q: How do I calculate the width of a room if I know the length and perimeter?
A: To calculate the width of a room, you can use the formula:
where $P$ is the perimeter, $l$ is the length, and $w$ is the width.
Q: What if the room has a different shape, such as a rectangle or a square?
A: If the room has a different shape, the formula for the perimeter will be different. For example, if the room is a square, the formula for the perimeter is:
where $P$ is the perimeter and $s$ is the side length.
Q: How do I calculate the perimeter of a room if I know the length and width?
A: To calculate the perimeter of a room, you can use the formula:
where $P$ is the perimeter, $l$ is the length, and $w$ is the width.
Q: What if the room has a different height?
A: If the room has a different height, the formula for the perimeter will also be different. However, the formula for the perimeter is not affected by the height of the room.
Q: How do I calculate the area of a room if I know the length and width?
A: To calculate the area of a room, you can use the formula:
where $A$ is the area, $l$ is the length, and $w$ is the width.
Q: What if I have a room with a different shape, such as a triangle or a circle?
A: If the room has a different shape, the formula for the area will be different. For example, if the room is a triangle, the formula for the area is:
where $A$ is the area, $b$ is the base, and $h$ is the height.
Q: How do I calculate the perimeter of a room if I know the area and length?
A: To calculate the perimeter of a room, you can use the formula:
where $P$ is the perimeter, $l$ is the length, and $A$ is the area.
Q: What if I have a room with a different perimeter?
A: If the room has a different perimeter, the formula for the width will also be different. You can use the formula:
where $P$ is the perimeter, $l$ is the length, and $w$ is the width.
Conclusion
In conclusion, this article has provided answers to frequently asked questions about the perimeter and area of a room. We have covered topics such as calculating the width of a room, calculating the perimeter of a room, and calculating the area of a room. We have also covered topics such as different shapes and formulas for different shapes.