In the cylinder, the radius is 5 cm, a section parallel to the axis is drawn, lagging behind it by a distance of 3 cm.

In the cylinder, the radius is 5 cm, a section parallel to the axis is drawn, lagging behind it by a distance of 3 cm.Find the height if the cross-sectional area is equal to 64 cm2

A cylinder is a body formed by rotating a rectangle around its side.
Since the section by a plane has the shape of a rectangle, its area is equal to the product of the width of this section by its length:
SAВСD = BC · AB.
Since the length of this section is equal to the height of the cylinder, then:
h = SAВСD / ВС.
To do this, it is necessary to find the length of the BC segment, which is the section width.
Consider the base of the cylinder.
Point O is its center, segments BO and OC are radii, as well as lateral sides of an isosceles triangle ΔBOC, segment BC is a secant plane, as well as the base of this triangle, segment OH is the distance from the secant plane to the axis, as well as the height of the triangle ΔВOС
Since the triangle ΔВOС is isosceles, then:
BH = HC;
ВС = ВН + НС.
Consider a triangle ΔBOН. This triangle is rectangular.
To calculate the ВН segment, we apply the Pythagorean theorem:
BO^2 = BH^2 + HO^2;
BH^2 = BO^2 – HO^2;
BH^2 = 5^2 – 3^2 = 25 – 9 = 16;
BH = √16 = 4 cm.
BC = 4 + 4 = 8 cm.
h= 64/8 = 8 cm.
Answer: The height of the cylinder is 8 cm.