What is the missing length of the triangular prism if the surface area is 524 square centimeters?
An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The sides and edges that are not visible are indicated by dashed lines. Parts are labeled with measurements. The left side of the triangle measures 10 centimeters. The right side of the triangle also measures 10 centimeters. The perpendicular height from the top vertex of the triangle to the base measures 6 centimeters. The base of the triangle measures 8 centimeters.
Let's try to find the missing length with the options provided (17 cm, 10 cm, 8 cm, 16 cm).
If we use 17 cm as the missing length, the surface area calculation would be as follows:
Area of the triangular face = 1/2 * 8 * 6 = 24 square cm
Area of the rectangular face = 10 * 17 = 170 square cm
Total surface area = 2(24) + 2(170) = 388 square cm
Since the given surface area is 524 cm^2, the missing length cannot be 17 cm.
We will repeat the calculations for the other provided options:
- 10 cm:
Area of the rectangular face = 10 * 10 = 100 square cm
Total surface area = 2(24) + 2(100) = 248 square cm
- 8 cm:
Area of the rectangular face = 10 * 8 = 80 square cm
Total surface area = 2(24) + 2(80) = 208 square cm
- 16 cm:
Area of the rectangular face = 10 * 16 = 160 square cm
Total surface area = 2(24) + 2(160) = 368 square cm
Comparing the calculations:
- The option 10 cm results in a total surface area of 248 cm^2.
- The option 8 cm results in a total surface area of 208 cm^2.
- The option 16 cm results in a total surface area of 368 cm^2.
Among the provided options, the missing length that results in a total surface area close to the given 524 cm^2 is 16 cm.
To find the missing length of the triangular prism, we first need to find the area of each face of the prism.
1. The area of the triangular face can be calculated using the formula for the area of a triangle: 1/2 * base * height.
Area of the triangular face = 1/2 * 8 * 6 = 24 square centimeters.
2. The area of the rectangular face can be calculated by multiplying the length and width of the face.
Area of the rectangular face = 10 * x = 10x square centimeters.
The total surface area of the prism is the sum of the areas of the two faces and the two identical side faces.
Total surface area = 2 * (Area of triangular face) + 2 * (Area of rectangular face)
524 = 2 * 24 + 2 * 10x
524 = 48 + 20x
476 = 20x
x = 23.8
Therefore, the missing length of the triangular prism is 23.8 centimeters.
these are the options though:
17 cm
10 cm
8 cm
16 cm
the dimensions of the triangular prism are:
Base=8
height=6
S1=10
S2=8
S3=10
find the missing length
To find the missing length of the triangular prism, we need to first calculate the lateral surface area of the prism and then use the total surface area provided to solve for the missing length.
Given:
Base = 8 cm
Height = 6 cm
Side 1 (S1) = 10 cm
Side 2 (S2) = 8 cm
Side 3 (S3) = 10 cm
Total Surface Area = 524 cm²
The lateral surface area of the triangular prism is the sum of the areas of the three rectangular faces:
Lateral Surface Area = (perimeter of the base) * height = (S1 + S2 + S3) * height
Substitute the given values:
Lateral Surface Area = (10 + 8 + 10) * 6
Lateral Surface Area = 28 * 6
Lateral Surface Area = 168 cm²
The total surface area of the prism includes the lateral surface area and the area of the two triangular faces on the ends:
Total Surface Area = Lateral Surface Area + 2 * (Area of one triangular face)
First, calculate the area of one triangular face:
Area of a Triangle = 0.5 * base * height
Area of one triangular face = 0.5 * 8 * 6
Area of one triangular face = 24 cm²
Now, calculate the missing length using the given Total Surface Area:
Total Surface Area = Lateral Surface Area + 2 * (Area of one triangular face)
524 = 168 + 2 * 24
524 = 168 + 48
524 = 216
To find the missing length:
216 = (S1 + S2 + S3) * height
216 = (10 + 8 + S3) * 6
216 = 18 + 6S3
6S3 = 216 - 18
6S3 = 198
S3 = 33
Therefore, the missing length of the triangular prism is 33 cm.