It is known that the bottom radius of a cone is 2cm and the height is 6cm. In all its inscribed cylinders, the maximum area of the cylinder side is calculated

The relation between radius R and height h of inscribed cylinder is (2-r): H = 2:6 H = 6-3r side area s = 2 π RH = 2 π R (6-3r) = 12 π R-6 π R. r = - 6 π (r.r-2r) s / - 6 π + 1 = R.R - 2R + 1 = (R-1) (R-1) because when (s / - 6 π + 1) has a minimum, s has a maximum, so when r = 1 (s / - 6 π + 1) has a minimum

The radius of the bottom surface of a cone is 2cm, the height is 6cm, and there is an inscribed cylinder with the height of X. 1) calculate the side area of the cone
2) When what is the value of X, the maximum value of the cylinder side is obtained

1) The bottom radius of the cone is 2cm and the height is 6cm,
Then the square root of the generatrix length of the cone is (6 ^ 2 + 2 ^ 2) = 40 and the square root of the cone is 2 √ 10
The arc length of the sector is 2 π * 2 = 4 π
Therefore, the side area of the cone is the area of the sector of the expanded side view of the cone: π * 4 π * 2 √ 10 = 8 √ 10 * π ^ 2 ()
2) Let the radius of the bottom circle of the inscribed cylinder with height X be r
There is R / 2 = (6-x) / 3
r=2-x/3 (1)
The side area of the cylinder is: 2 π R * x = 2 π (2-x / 3) * x = - (2 π / 3) x ^ 2 + 4 π X
=-(2π/3)(x-3)^2+6π
When x = 3cm
The maximum side area of the cylinder is 6 π square centimeter

If the volume ratio of cone and cylinder is 3:1, are these two figures of equal height and equal base?
As shown in the picture

"Kexinweier":
The volume ratio of a cone to a cylinder is 1:3 (not 3:1). These two figures (not necessarily) have the same base and height
If the diameter of the bottom of the cone is n & sup2; times of the diameter of the cylinder, and the height of the cone is 1 / N of the height of the cylinder, then their volume ratio is still 1:3
Good luck and goodbye

As shown in the figure, △ ABC is surrounded by rubber bands to fix the BC side
(1) If you pull point a vertically upward, what will happen to the size of ∠ a, ∠ B, ∠ C? If you move it downward?
(2) If ∠ a decreases degree a, B increases degree B and C increases degree C in the process of motion, what is the relationship between ABC

[reference answer]

(1) Fixed BC side, pull point a upward, then ∠ a decreases, and ∠ B and ∠ C increase
&The effect is as follows:

If a is pulled down, a increases, B and C decrease
The effect is as follows:
(2) Relation: a = B + C
According to the fact that the sum of the internal angles of the triangle is 180 degrees, the three internal angles of the triangle should also be 180 degrees
(∠A-a)+(∠B+b)+(∠C+c)=180°
(∠A+∠B+∠C)-(a-b-c)=180°
180°-(a-b-c)=180°
So a-b-c = 0
∴a=b+c

If the radius of the bottom of a cone is 3 cm and the height is 4 cm, the total area of the cone is -------- cm square

Side: π * 3 * radical (3 ^ 2 + 4 ^ 2) = 15 π
Bottom: π * 3 ^ 2 = 9 π
Total: 15 π + 9 π = 24 π cm ^ 2

As shown in the figure, AB and CD are two parallel wooden strips fixed on the board. Fix a rubber band at points a and C, and point E is the rubber band. After pulling point e to tighten the rubber band, please explore the relationship between ∠ a, ∠ C and ∠ AEC, and explain the reason

The sum of the three angles is 360 degrees
Because, the sum of the inner angles of the Pentagon ABCDE is 540 degrees, because it is parallel, so the sum of the angles B and D is 180 degrees, and the remaining sum is 360 degrees!

The radius of the bottom circle is 3cm and the side area of the cone with the height of 4cm is ()
A. 7.5πcm2B. 12πcm2C. 15πcm2D. 24πcm2

∵ the radius of the bottom circle is 3cm, the height is 4cm, the generatrix length of the cone is 5cm, and the side area of the cone = π × 3 × 5 = 15 π cm2

The focus of passing parabola Y2; = 4x
What is the area of △ poq if we make a straight line intersection parabola with an inclination of 3 π / 4 at two points P and Q, and O is the origin of the coordinate?

Y ^ 2 = 2 * 2x, P = 2, focus f (1,0), linear equation is y = - (x-1), x + Y-1 = 0, let P (x1, Y1), q (X2, Y2), substitute into parabolic equation, x ^ 2-2x + 1 = 4x, x ^ 2-6x + 1 = 0, according to Veda theorem, X1 + x2 = 6, x1x2 = 1, according to chord length formula, | PQ | = √ (1 + 1) (x1-x2) ^ 2 = √ 2 * [X1 + x2) ^ 2-4x1x2]

As shown in the figure, if the bottom radius of the cone is 6cm and the height is 8cm, then the side area of the cone is______ cm2．

If the radius of the bottom is 6cm and the height is 8cm, then the perimeter of the bottom is 12 π. According to the Pythagorean theorem, if the length of the generatrix is 10, then the side area is 12 × 12 π × 10 = 60 π cm2

It is known that the parabola y ^ 2 = 2px (P > 0), the intersection parabola of the straight line passing through the focus f lies at two points m and N, and the minimum area of ⊿ mon is 1 / 2, where o is the origin of the coordinate
(1) Solving the equation of parabola;
(2) Through point a (- P / 2,0), make a straight line complementary to the inclination angle of straight line Mn, intersect the parabola at two points B and C, prove that: (∣ ab ∣. ∣ AC ∣) / (∣ FM ∣. ∣ FN ∣) is the fixed value, and calculate the fixed value

(1) Let the linear equation be x = my + P / 2 and Y ^ 2 = 2px, then we obtain the Weida theorem of Y ^ 2-2pmy-p ^ 2 = 0, which is transformed into | y1-y2 | = 2p * root (m ^ 2 + 1) s ⊿ mon = P ^ 2 / 2 * (M ^ 2 + 1), so when m = 0, we get the minimum value, that is, P ^ 2 / 2 = 1 / 2, the solution is p = 1, the parabolic equation y ^ 2 = 2x (2) is complementary, then the slopes are mutual

It is known that the bottom radius of a cone is 2cm and the height is 6cm. In all its inscribed cylinders, the maximum area of the cylinder side is calculated