There is a cylinder 10 cm high. If its height is reduced by 2 cm, its surface area will be reduced by 18.84 square cm. What is the volume of the original cylinder I need it before 8 o'clock

There is a cylinder 10 cm high. If its height is reduced by 2 cm, its surface area will be reduced by 18.84 square cm. What is the volume of the original cylinder I need it before 8 o'clock

18.84 = 2 * 3.14 * r (radius of bottom circle) * 2
r=1.5cm
V = 3.14 * R2 (square) * H (10cm) = 70.65

Calculate ∫ ∫ (Z + 2x + 4 / 3Y) ds, where ∑ is the part of plane X / 2 + Y / 3 + Z / 4 = 1 in the first hexagram

The two sides of the plane equation are multiplied by 4 to get Z + 2x + 4 / 3Y = 4, so the integral ∫ (Z + 2x + 4 / 3Y) ds = ∫ - 4ds. Next, the area of △ enclosed by the three intersections of the plane and the three coordinate axis can be calculated. For example, the volume of the tetrahedron can be calculated, and the distance from the origin to the plane can be calculated, so the area of the triangle can be calculated
The curve area can also be divided into double integrals, and the partial derivatives of Z with respect to X and y can be obtained. DS = √ (61) / 3dxdy. The projection region of ∑ on the xoy plane is bounded by x = 0, y = 0, X  2 + y  3 = 1
So ∫ (Z + 2x + 4 / 3Y) ds = ∫ ∫ 4ds = ∫ ∫ 4 ×√ (61) / 3dxdy = 4 ×√ (61) / 3 × 1 / 2 × 2 × 3 = 4 √ (61)

Let ∑ be the part of plane x + y + Z = 1 in the first hexagram, then ∫ 6 (2x + y + Z + 1) DXDY is equal to
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The original formula = 6 ∫ DX ∫ (2x + y + (1-x-y) + 1) dy (∫ x + y + Z = 1, reduced by graphic analysis) = 6 ∫ DX ∫ (x + 2) dy = 6 ∫ (x + 2) (1-x) DX = 6 ∫ (2-x-x & # 178;) DX = 6 (2-1 / 2-1 / 3) = 7

Let ∑ be the plane X / 2 + Y / 3 + Z / 4 = 1, then ∫ (Z + 2x + 4 / 3Y) ds =

∑：0

If 2x + 3y-4z = 7, x + 2y-5z = 2, then the value of X + y + Z is

2X + 3y-4z = 7
X + 2y-5z = 2 is a bivariate
In duplicate minus two
We get x + y + Z = 5

2X + 3y-4z = 7, x + 3y-5z = 2, then the value of X + 2y-3z is ()
2X + 3y-4z = 7, x + 3y-5z = 2, then the value of X + 2y-3z is

2x+3y-4z=7,
x+3y-5z=2
Add
3x+6y-9z=9
Divide two sides by three
x+2y-3z=3

What is the eccentricity of an ellipse if the cross section of a plane cylinder at an angle of 30 ° to the bottom is an ellipse?

One half
If the cylinder is cut by a plane at an angle of 30 ° to the bottom, the length of the major axis of the ellipse is
2a=2R/cos30°
The minor axis length is 2B = 2R
So the major half axis of the ellipse a = R / cos30 ° = 2R / √ 3
Short half axis B = R
So the focal length C = √ (a ^ 2-B ^ 2) = R / √ 3
So the centrifugal ratio is e = C / a = 1 / 2

It is known that the diameter of the bottom of the cylinder is 2R, and a plane with an angle of 30 ° to the bottom cuts the cylinder, then the eccentricity of the ellipse on the section is 0_______ .

The length of the major axis of the ellipse cut at an angle of 30 ° is 2R / cos 30 ° and the length of the minor axis is 2R, so a = 2 3 K / 3, B = R.C = 3 / 3, e = 1 / 2

As shown in the figure, a cylinder with a diameter of 20 on the bottom is cut by a plane with a dihedral angle of 60 ° with the bottom, and the cross section is an ellipse, then the focal length of the ellipse is______ ．

It can be seen from the meaning that the short axis length of the ellipse is 20 ∵ the cylinder with a diameter of 20 at the bottom is cut by a plane with a dihedral angle of 60 ° with the bottom. The cross section of the ellipse is an ellipse. The figure of the cross section of the ellipse through the long axis is shown in the figure, ∠ KJL = 60 ° and JK is the diameter of the bottom and the length of the ellipse is 20. Therefore, the triangle is a right triangle, so LJ = 40 ∵ the long axis length of the ellipse

As shown in the figure, a cylinder with a diameter of 12 cm on the bottom is cut by a plane at 30 ° to the bottom. If the section is an ellipse, the eccentricity of the ellipse is 0___ ．

Because the cylinder with a diameter of 12 cm on the bottom is cut by a plane which is 30 ° to the bottom, and its section is an ellipse, the short half axis of the ellipse is 6, the long half axis is 6cos, 30 ° = 43, ∵ A2 = B2 + C2, ∵ C = 23, ∵ the eccentricity of the ellipse is e = CA = 2343 = 12