Coordinate Geometry and Nets Unit Test
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Question
Use the image to answer the question.
An illustration shows the net of a triangular prism. The illustration forms a shape where the small triangle is in the middle with each side being the base of three different triangles. The height of the triangle in the middle is 5 inches as shown by a dashed line and the base is 6 inches.
Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.2 .
(1 point)
inches
To find the slant height of the triangular pyramid made of equilateral triangles, we first need to find the lateral area of one of the three triangles on the net.
The lateral area of an equilateral triangle can be calculated using the formula:
Lateral Area = (s^2 * √3) / 4
where s is the length of the side of the equilateral triangle.
Since the base of the triangular prism is 6 inches, the side length of the equilateral triangle is also 6 inches.
Plugging in the values:
Lateral Area = (6^2 * √3) / 4
Lateral Area = (36 * √3) / 4
Lateral Area = 9√3
Since there are three congruent triangles on the net, the total lateral area is 3 times the lateral area of one triangle:
Total Lateral Area = 3 * 9√3
Total Lateral Area = 27√3
Given that the surface area of the triangular prism is 78 in^2, we know that the total lateral area plus the area of the two triangular bases is equal to 78:
27√3 + 2(6 * 5 / 2) = 78
27√3 + 30 = 78
27√3 = 48
√3 = 48 / 27
√3 = 16 / 9
√3 = 4 / 3
Therefore, the slant height of the triangular pyramid is 4 inches.