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To determine the surface area of the solid formed by placing a right triangular prism atop a cube, we need to follow several steps—which include calculating the surface areas of both the triangular prism and the cube, and then accounting for the overlap where they meet.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions to facilitate our calculations:
- Height of the prism: [tex]\(1 \frac{5}{6}\)[/tex] inches.
[tex]\[ 1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6} \approx 1.8333 \text{ inches} \][/tex]
- Height of the triangular base: [tex]\(8 \frac{2}{3}\)[/tex] inches.
[tex]\[ 8 \frac{2}{3} = 8 + \frac{2}{3} = \frac{24}{3} + \frac{2}{3} = \frac{26}{3} \approx 8.6667 \text{ inches} \][/tex]
### Step 2: Calculate the Area of the Triangular Base
Using the formula for the area of a triangle, [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 10 \times 8.6667 = \frac{1}{2} \times 86.6667 = 43.3333 \text{ square inches} \][/tex]
### Step 3: Calculate the Lateral Surface Area of the Triangular Prism
The prism has three rectangular faces, each with a length of 10 inches and a height equal to the height of the prism [tex]\(1 \frac{5}{6}\)[/tex] inches:
[tex]\[ \text{Lateral Surface Area} = 3 \times 10 \times 1.8333 = 55.0000 \text{ square inches} \][/tex]
### Step 4: Calculate the Total Surface Area of the Triangular Prism
The total surface area includes both triangular bases and the lateral surface area:
[tex]\[ \text{Total Surface Area of the Prism} = 2 \times 43.3333 + 55.0000 = 141.6667 \text{ square inches} \][/tex]
### Step 5: Calculate the Surface Area of the Cube
Each side of the cube measures 10 inches, and the cube has 6 faces:
[tex]\[ \text{Surface Area of the Cube} = 6 \times (10 \times 10) = 6 \times 100 = 600 \text{ square inches} \][/tex]
### Step 6: Consider the Overlapping Area
The triangular base of the prism completely covers one face of the cube. Thus, this overlapping area needs to be subtracted from the total surface area once:
[tex]\[ \text{Overlapped Area} = 43.3333 \text{ square inches} \][/tex]
### Step 7: Calculate the Surface Area of the Solid
Finally, we combine the surface areas of the prism and the cube, then subtract the overlapping area:
[tex]\[ \text{Surface Area of the Solid} = 600 + 141.6667 - 43.3333 = 698.3334 \text{ square inches} \][/tex]
Therefore, the surface area of the solid formed is approximately [tex]\( 698.3333 \)[/tex] square inches.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions to facilitate our calculations:
- Height of the prism: [tex]\(1 \frac{5}{6}\)[/tex] inches.
[tex]\[ 1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6} \approx 1.8333 \text{ inches} \][/tex]
- Height of the triangular base: [tex]\(8 \frac{2}{3}\)[/tex] inches.
[tex]\[ 8 \frac{2}{3} = 8 + \frac{2}{3} = \frac{24}{3} + \frac{2}{3} = \frac{26}{3} \approx 8.6667 \text{ inches} \][/tex]
### Step 2: Calculate the Area of the Triangular Base
Using the formula for the area of a triangle, [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times 10 \times 8.6667 = \frac{1}{2} \times 86.6667 = 43.3333 \text{ square inches} \][/tex]
### Step 3: Calculate the Lateral Surface Area of the Triangular Prism
The prism has three rectangular faces, each with a length of 10 inches and a height equal to the height of the prism [tex]\(1 \frac{5}{6}\)[/tex] inches:
[tex]\[ \text{Lateral Surface Area} = 3 \times 10 \times 1.8333 = 55.0000 \text{ square inches} \][/tex]
### Step 4: Calculate the Total Surface Area of the Triangular Prism
The total surface area includes both triangular bases and the lateral surface area:
[tex]\[ \text{Total Surface Area of the Prism} = 2 \times 43.3333 + 55.0000 = 141.6667 \text{ square inches} \][/tex]
### Step 5: Calculate the Surface Area of the Cube
Each side of the cube measures 10 inches, and the cube has 6 faces:
[tex]\[ \text{Surface Area of the Cube} = 6 \times (10 \times 10) = 6 \times 100 = 600 \text{ square inches} \][/tex]
### Step 6: Consider the Overlapping Area
The triangular base of the prism completely covers one face of the cube. Thus, this overlapping area needs to be subtracted from the total surface area once:
[tex]\[ \text{Overlapped Area} = 43.3333 \text{ square inches} \][/tex]
### Step 7: Calculate the Surface Area of the Solid
Finally, we combine the surface areas of the prism and the cube, then subtract the overlapping area:
[tex]\[ \text{Surface Area of the Solid} = 600 + 141.6667 - 43.3333 = 698.3334 \text{ square inches} \][/tex]
Therefore, the surface area of the solid formed is approximately [tex]\( 698.3333 \)[/tex] square inches.
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