A cone is inscribed in a cylinder with a height of 10 (the base of the cone coincides with the lower base of the cylinder

| February 23, 2021 | education

A cone is inscribed in a cylinder with a height of 10 (the base of the cone coincides with the lower base of the cylinder, the top of the cone is the middle of the upper base of the cylinder), the angle between the intersecting generatrices of the cylinder and the cone is 30 degrees. Find the area of the entire surface of the cone

In a right-angled triangle ACO1, we determine the length of the leg CO1, which is the radius of the cone and cylinder.

tg30 = CO1 / AC.

CO1 = AC * tg30 = 10 * 1 / √3 = 10 * √3 / 3 cm.

Let us determine the length of the hypotenuse O1A.

Leg CO1 lies against an angle of 300, then the length of the hypotenuse is equal to two lengths of this leg.

AO1 = 2 * CO1 = 2 * 10 * √3 / 3 = 20 * √3 / 3 cm.

Determine the area of ​​the base of the cone.

Sop = n * CO12 = n * (10 * √3 / 3) 2 = n * 300/9 = n * 100/3 cm2.

Let us determine the area of ​​the lateral surface of the cone.

Side = n * CO1 * AO1 = n * (10 * √3 / 3) * (20 * √3 / 3) = n * 600/9 = n * 200/3 cm2.

Then Spol = Sb + Sbok = n * 100/3 + n * 200/3 = n * 100 cm2.

Answer: The total surface area of ​​the cone is n * 100 cm2.



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