Question

A solid is formed by a pyramid mounted on a cube, the base coinciding with cube's upper face. Side of cube is `16 cm` and the apothem of the pyramid is `17 cm`. Calculate: the total surface area, the volume and mass knowing is made wood `0.65 g/(cm^3)`?

Answer 1

Total Surface Area: `\ color(green)(T S A = "1824 cm"^2)`

Volume: `\ color(maroon)(V_t = V_c + V_p = "23296 cm"^3)`

Mass of wood: `\ color(blue)(m = d * V_t = 15,142.4 \ "g")`

Explanation:

Side `= a = "16 cm"`, apothem `= l = "17 cm"`

Total Surface Area = Base Area of Cube + Lateral Surface Area of the Cube + Surface Area of Pyramid

Base Area of Cube `A_b = a^2 = "256 cm"^2`

Lateral Surface Area of Cube `A_c = 4 a^2 = "1024 cm"^2`

Surface Area of Pyramid `= A_p = 4 * (a/2) * l = 4 * 8 * 17 = "544 cm"^2`

`T S A = 256 + 1024 + 544 = "1824 cm"^2`

`"Total Volume" \ (V_t) = "Volume of Cube" \ (V_c) + "Volume of Pyramid" \ (V_p)`

`V_c = a^3 = 16^3 = "4096 cm"^3`

`V_p = (1/3) * a^2 * h = (1/3) * a^2 * sqrt(l^2 - (a/2)^2)`

`V_p = (1/3) * 16^2 * sqrt(17^2 - 8^2)`

`V_p = (1/3) * 256 * 225 = "19200 cm"^3`

`"Volume" \ V_t = V_c + V_p = 4096 + 19200 = "23296 cm"^3`

Mass of Wood `m = "density" * "volume"`

`m = d * V_t = 0.65 * 23296 = 15,142.4` `"g"`