Question
L shape platform
16 inches length on top 6 inches down right side, 10 inches down left side the little square L part is 4 across and 4 inches towards the 12 inch length
What is the surface area of the platform if it is 3 inches thick
16 inches length on top 6 inches down right side, 10 inches down left side the little square L part is 4 across and 4 inches towards the 12 inch length
What is the surface area of the platform if it is 3 inches thick
Answers
GPT-4o mini
To calculate the surface area of the L-shaped platform, we can break it down into its main components and then sum their areas.
### Step 1: Understand the dimensions
You described the L-shape platform as having the following dimensions:
- The top part is 16 inches long.
- The right side down is 6 inches.
- The left side down is 10 inches.
- There is a 4-inch by 4-inch square cut-out from the inside corner of the L.
Since the platform is 3 inches thick, we will calculate the surface area excluding the cut-out and then incorporate the thickness.
### Step 2: Calculate the area of the outer dimension
1. **Width and Height of the outer L shape:**
- The width of the platform is 16 inches.
- The height extends down to 10 inches on the left and 6 inches on the right.
To find the total area of the outer rectangle:
\[
\text{Outer area} = \text{length} \times \text{height}
\]
However, we calculate an approximate rectangle for the L shape:
- Height: The maximum height (on the left) is 10 inches,
- Width: The maximum width (full length) is 16 inches.
Thus, the **outer rectangle area** is:
\[
\text{Outer Area} = 16 \, \text{inches} \times 10 \, \text{inches} = 160 \, \text{square inches}
\]
### Step 3: Calculate the area of the inner cut-out
2. **Area of the cut-out:**
- The cut-out is a 4-inch by 4-inch square.
So, the **inner cut-out area** is:
\[
\text{Cut-out Area} = 4 \, \text{inches} \times 4 \, \text{inches} = 16 \, \text{square inches}
\]
### Step 4: Calculate the total surface area
To find the adjusted surface area of the L shape, we subtract the area of the cut-out from the outer area:
\[
\text{Surface area of the L-shape} = \text{Outer Area} - \text{Cut-out Area}
\]
\[
\text{Surface area of the L-shape} = 160 \, \text{square inches} - 16 \, \text{square inches} = 144 \, \text{square inches}
\]
### Step 5: Adding the thickness
Since the platform is 3 inches thick, we must consider the sides. The primary exposed areas will be the top, the perimeter sides, and the bottom.
1. **Top area** is the surface area calculated, which is \(144 \, \text{square inches}\).
2. **Perimeter surface area** (the vertical sides):
- Two long sides (where the length is 16 inches and height is 3 inches) and two shorter sides (the heights of 6 inches and 10 inches with a width of 3 inches).
#### Calculate side areas:
- For the **two longer vertical sides** (16 inches length):
\[
2 \times (16 \, \text{inches} \times 3 \, \text{inches}) = 2 \times 48 \, \text{square inches} = 96 \, \text{square inches}
\]
- For the **two shorter vertical sides** (10 inches and 6 inches heights):
- Short right side (6 inches):
\[
1 \times (3 \, \text{inches} \times 6 \, \text{inches}) = 18 \, \text{square inches}
\]
- Short left side (10 inches):
\[
1 \times (3 \, \text{inches} \times 10 \, \text{inches}) = 30 \, \text{square inches}
\]
So:
\[
\text{Total side area} = 96 + 18 + 30 = 144 \, \text{square inches}
\]
### Final Surface Area Calculation
The total surface area therefore is:
\[
\text{Total Surface Area} = \text{Top Area} + \text{Side Areas} + \text{Bottom Area (equal to top area)}
\]
Putting it together:
\[
\text{Total Surface Area} = 144 + 144 + 144 = 432 \, \text{square inches}
\]
### Final Result
Thus, the surface area of the L-shaped platform is **432 square inches**.
### Step 1: Understand the dimensions
You described the L-shape platform as having the following dimensions:
- The top part is 16 inches long.
- The right side down is 6 inches.
- The left side down is 10 inches.
- There is a 4-inch by 4-inch square cut-out from the inside corner of the L.
Since the platform is 3 inches thick, we will calculate the surface area excluding the cut-out and then incorporate the thickness.
### Step 2: Calculate the area of the outer dimension
1. **Width and Height of the outer L shape:**
- The width of the platform is 16 inches.
- The height extends down to 10 inches on the left and 6 inches on the right.
To find the total area of the outer rectangle:
\[
\text{Outer area} = \text{length} \times \text{height}
\]
However, we calculate an approximate rectangle for the L shape:
- Height: The maximum height (on the left) is 10 inches,
- Width: The maximum width (full length) is 16 inches.
Thus, the **outer rectangle area** is:
\[
\text{Outer Area} = 16 \, \text{inches} \times 10 \, \text{inches} = 160 \, \text{square inches}
\]
### Step 3: Calculate the area of the inner cut-out
2. **Area of the cut-out:**
- The cut-out is a 4-inch by 4-inch square.
So, the **inner cut-out area** is:
\[
\text{Cut-out Area} = 4 \, \text{inches} \times 4 \, \text{inches} = 16 \, \text{square inches}
\]
### Step 4: Calculate the total surface area
To find the adjusted surface area of the L shape, we subtract the area of the cut-out from the outer area:
\[
\text{Surface area of the L-shape} = \text{Outer Area} - \text{Cut-out Area}
\]
\[
\text{Surface area of the L-shape} = 160 \, \text{square inches} - 16 \, \text{square inches} = 144 \, \text{square inches}
\]
### Step 5: Adding the thickness
Since the platform is 3 inches thick, we must consider the sides. The primary exposed areas will be the top, the perimeter sides, and the bottom.
1. **Top area** is the surface area calculated, which is \(144 \, \text{square inches}\).
2. **Perimeter surface area** (the vertical sides):
- Two long sides (where the length is 16 inches and height is 3 inches) and two shorter sides (the heights of 6 inches and 10 inches with a width of 3 inches).
#### Calculate side areas:
- For the **two longer vertical sides** (16 inches length):
\[
2 \times (16 \, \text{inches} \times 3 \, \text{inches}) = 2 \times 48 \, \text{square inches} = 96 \, \text{square inches}
\]
- For the **two shorter vertical sides** (10 inches and 6 inches heights):
- Short right side (6 inches):
\[
1 \times (3 \, \text{inches} \times 6 \, \text{inches}) = 18 \, \text{square inches}
\]
- Short left side (10 inches):
\[
1 \times (3 \, \text{inches} \times 10 \, \text{inches}) = 30 \, \text{square inches}
\]
So:
\[
\text{Total side area} = 96 + 18 + 30 = 144 \, \text{square inches}
\]
### Final Surface Area Calculation
The total surface area therefore is:
\[
\text{Total Surface Area} = \text{Top Area} + \text{Side Areas} + \text{Bottom Area (equal to top area)}
\]
Putting it together:
\[
\text{Total Surface Area} = 144 + 144 + 144 = 432 \, \text{square inches}
\]
### Final Result
Thus, the surface area of the L-shaped platform is **432 square inches**.