Question

Vignesh owns a cottage in the shape of a cube with each edge of length 26 feet. The roof is in the shape of a square pyramid and it extends two feet over the edge of the cottage on each side. The lateral sides of the roof are 17 feet long. What is the total surface area of the roof?

Answer 1

Answer:

Step-by-step explanation:

The shape of the cottage is cube

The edge of the cube is of length is 26ft

L_1 = 26ft.

The roof is in form of a square with with length 17ft.

L_2 = 17ft.

The roof is made up of 4 congruent isosceles triangles. Since the roof extends 2 ft over the edge of the cottage on each side, then the base of each triangle is 26 + 2 = 28 ft.

L_2 = 28 ft

Then, area of the roof is

A = 28²

A = 784 ft²

The triangular pyramid.

The triangular has four sides

Then,

Each of the area can be calculated using

A = ½ × b × h

Then, the four area of the triangular pyramid

A_total = 4 × ½ × b × h

A_total = 2 × b × h

The base of the triangle is 28ft.

The calculate the height of the triangle

Let's calculate the area of one of the triangles and then just multiply by 4. See attachment. Drop a perpendicular from the vertex of a triangle to its base. We now have the triangle broken up into two right triangles. The hypotenuse is 17 ft and one of the legs of the right triangle is 28 / 2 = 14 ft. Then the height is using Pythagoras theorem

a² = b² + c²

17² = 14² + h²

17² - 14² = h²

h² = 93

h = √93

Then, the area of the triangular pyramid is

A = 2 × b × h

A = 2 × 28 × √93

A = 540.04 ft²

Then, the area of the pyramid is approximately 540 ft², since the base of the pyramid cannot be seen.

But if we include the base, then, the total surface area is

A_t = A_triangular + A_base

A_t = 540 + 784

A_t = 1324 ft²