The rectangle with the length of 4cm and the width of 2cm respectively revolves around the straight line where the length and the width lie to get two geometric bodies. Which geometric body has the largest volume and the surface area? Drawing calculation instructions

The rectangle with the length of 4cm and the width of 2cm respectively revolves around the straight line where the length and the width lie to get two geometric bodies. Which geometric body has the largest volume and the surface area? Drawing calculation instructions

1. A cylinder is obtained by rotating around the length, then
The radius of the bottom surface is 2cm, the height is 4cm, and the volume v = π × 2 × 2 × 4 = 16 π cm3
Surface area: 2 * 3.14 * 2 ^ 2 + 2 * 3.14 * 2 * 4
=6.28*12
=75.36 cm2
2. A cylinder is obtained by rotating around the width,
The radius of the bottom is 4cm, the height is 2cm, and the volume v = π × 4 × 2 = 32 π cm3
Surface area: 2 * 3.14 * 4 ^ 2 + 2 * 3.14 * 4 * 2
=6.28*24
=150.72cm2
That is to say, a cylinder with large volume and surface area can be obtained by rotating around the width

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What is the surface area and volume of a geometric figure formed by a rectangle with a length of 5cm and a width of 4cm rotating around one side?

(1)
When the length of the rectangle is taken as the axis
The base area is π * 4 & sup2; = 16 π
Side area = bottom perimeter × height = 2 π * 4 * 5 = 40 π
The surface area is 40 π + 2 × 16 π = 72 π ≈ 226.08 CM & sup2;
Volume = bottom area × height = 16 π × 5 = 80 π≈ 251.2cm & sup3;
(2)
When the width of the rectangle is taken as the axis
The base area is π * 5 & sup2; = 25 π
Side area = bottom perimeter × height = 2 π * 5 * 4 = 40 π
The surface area is 40 π + 2 × 25 π = 90 π ≈ 282.6cm & sup2;
Volume = bottom area × height = 25 π × 4 = 100 π≈ 314cm & sup3;

If a right triangle is 4, 3 and 5 long respectively, and rotated one circle with the longest side as the axis, what is the volume of the figure?
The PI is 3
emergency

Cut along the height of the hypotenuse, one cone above and one cone below
The height (i.e. the radius of the bottom of the cone) is 12 / 5
The volume sum of two cones is
(12/5)^2*3*5/3=144/5

The length of the three sides of a right triangle is 3, 4 and 5. If each side is rotated for one circle, three solids are obtained. The ratio of the largest volume to the smallest volume of the three solids is calculated

(1) The volume of a cone is V3 = 13 π × 42 × 3 = 16 π when the right angle side with length of 3 is divided into axes; (2) the solid with length of 4 as axis is also a cone, and the volume is V4 = 13 π × 32 × 4 = 12 π; (3) the solid with length of 5 as axis is a spinning cone composed of two cones whose bottom surfaces are superposed up and down Let the radius of the bottom be h, which is the height of the hypotenuse of the right triangle. According to the area formula of the right triangle: 12 × 5h = 12 × 3 × 4, so h = 125. Then the volume of the spinning cone is calculated by the volume formula of the cone: V5 = 13 π h2h1 + 13 π h2h2 = 13 π H2 (H1 + H2) = 13 π (125) 2 × 5 = 485 π; (4) 16 π > 12 π > 485 π, 16 π: 485 π = 5:3 The smallest volume ratio is 5:3