The tower consists of a regular four-cornered prism and a regular four-cornered pyramid, the base of the prism

univerkov | March 28, 2021 | education

The tower consists of a regular four-cornered prism and a regular four-cornered pyramid, the base of the prism is equal to the base of the pyramid. The lateral edges of the prism are perpendicular to the base of the prism. The height of the prism is 3 times the height of the pyramid. Apothem of the pyramid 40 cm. The length of the lateral rib is 50 cm. Find the surface area of the tower.

Let’s start with a pyramid. In a right-angled triangle ОНD1, the hypotenuse ОD1 = 50 cm, and the leg ОН = 40 cm, then, according to the Pythagorean theorem, НD1 ^ 2 = ОD1 ^ 2 – ОН ^ 2 = 2500 – 1600 = 900.

НД1 = 30 cm.

Since there is a square at the base of the pyramid, then НD1 = НК = 30 cm, and D1С1 = AB = BC = CD = AD = 2 * НD1 = 60 cm.

From the right-angled triangle OKН, according to the Pythagorean theorem, we determine the height of the pyramid OK.

OK ^ 2 = OH ^ 2 – KH ^ 2 = 1600 – 900 = 700.

OK = 10 * √7 cm.

By condition, the height of the prism is three times the height of the pyramid, then KP = 3 * OK = 3 * 10 * √7 = 30 * √7 cm.

Determine the surface area of the prism.

Spr = Sbok + Sbn = 4 * AB * KР * AB * AD = 3 * 60 * 30 * √7 + 60 * 60 = 5400 * √7 + 3600 cm2.

Determine the surface area of the pyramid.

Spir = 4 * OH * D1C1 / 2 = 4 * 40 * 60/2 = 1200 cm2.

Determine the surface area of the tower.

S = Spr + Spir = 5400 * √7 + 3600 + 1200 = 6600 * √7 + 3600 = 600 * (11 * √7 + 6) cm2.

Answer: The surface area of the tower is 600 * (11 * √7 + 6) cm2.

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