# The height of the cylinder is 5√3 cm, and the diagonal of the axial section makes an angle of 30 ° with the base plane.

The height of the cylinder is 5√3 cm, and the diagonal of the axial section makes an angle of 30 ° with the base plane. Find the volume of the cylinder.

The axial section of the cylinder is a rectangle, two sides of which are equal to the height of the cylinder, the other two sides are equal to the diameter of the base of the cylinder.
Consider a right-angled triangle, in which the legs are the height of the cylinder and the diameter of the base, the hypotenuse is the diagonal of the axial section. Since the diagonal of the axial section forms an angle of 30 degrees with the base plane, the leg adjacent to this angle is the diameter of the base, the opposite leg is the height of the cylinder. The ratio of the opposite leg to the adjacent leg is equal to the tangent of the angle. Means: tg30 = H / d. Hence d = H / tg30 = (5√3) / (√3 / 3) = 15 cm.
The volume of a cylinder is equal to the product of the base area by the height:
V = Sbas * H = π * (d ^ 2/4) * H = π * (15 ^ 2/4) * 5√3 = π * 225 * 5√3 / 4, which is approximately equal to 1530.39 cm2.

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