OXYZ is a tetrahedron whose faces are identical equilateral triangles of side 10 m each. Given that the slant height of pyramid is 5 square root 3 metres

find its total surface area leaving your answer correct to 3 significant figures.

To find the total surface area of the tetrahedron, we need to calculate the area of each face and sum them up.

1. First, let's find the height of the tetrahedron. Since the faces are equilateral triangles, we can use the Pythagorean theorem to find the height 'h' of one triangle:
h = (√3/2) * side
h = (√3/2) * 10 m
h = 5√3 m

2. The slant height 'l' of the pyramid is given as 5√3 m.

3. Using the slant height and the height, we can find the length of the altitude 'a' from one of the vertices of the tetrahedron to the center of its opposite face:
a = (√3/3) * l
a = (√3/3) * 5√3 m
a = (3/3) * 5 m
a = 5 m

4. Each face of the tetrahedron is an equilateral triangle of side length 10 m. Using the formula for finding the area of an equilateral triangle:
Area = (square root of 3 / 4) * side^2
Area = (√3 / 4) * (10 m)^2
Area = (√3 / 4) * 100 m^2
Area = 25√3 m^2

5. Since OXYZ is a tetrahedron with four identical faces, the total surface area is four times the area of one face.
Total Surface Area = 4 * 25√3 m^2
Total Surface Area = 100√3 m^2

Therefore, the total surface area of the tetrahedron OXYZ is 100√3 square meters.

To find the total surface area of the tetrahedron, we need to calculate the areas of all its faces and add them together.

First, let's find the area of one equilateral triangle face.

The area of an equilateral triangle can be calculated using the formula:

Area = (sqrt(3) / 4) * (side)^2

Given that each side of the triangle is 10 m, the area of one equilateral triangle face is:

Area of one face = (sqrt(3) / 4) * (10)^2
= (sqrt(3) / 4) * 100
= 25 * sqrt(3) m^2

Next, we need to calculate the area of the triangular face formed by the three slant heights.

The slant height (l) of the tetrahedron is given as 5 * sqrt(3) m. To find the height (h) of the triangle, we can divide the slant height by 2:

h = (l / 2)
= (5 * sqrt(3)) / 2
= (5/2) * sqrt(3)

Now let's find the base (b) of the triangle. The base is the side length of the equilateral triangle, which is given as 10 m.

Using the formula for the area of a triangle:

Area of triangular face = (1/2) * b * h
= (1/2) * 10 * ((5/2) * sqrt(3))
= 25/2 * sqrt(3) m^2

Since there are four triangular faces in a tetrahedron, the total area of the triangular faces is:

Total area of triangular faces = 4 * (Area of triangular face)
= 4 * (25/2 * sqrt(3))
= 50 * sqrt(3) m^2

Finally, to find the total surface area, we need to add the area of the equilateral triangle faces to the area of the triangular faces:

Total surface area = Area of equilateral triangle faces + Total area of triangular faces
= 4 * (25 * sqrt(3)) + 50 * sqrt(3) m^2
= 100 * sqrt(3) + 50 * sqrt(3) m^2
= 150 * sqrt(3) m^2

Therefore, the total surface area of the tetrahedron is approximately 259.807 m^2 (rounded to 3 significant figures).

All that we really needed to know was that the sides were 10 m each for each of the 4 triangles.

We could have found the height using Pythagoras.

Area of one of them = (1/2)base x height = (1/2)(10)(5√3)
So four of them will have an area of
4(1/2)(10)(5√3) m^2 = 100√3 m^2 or appr .....