As shown in the figure, the side of the cone is cut along OA, and the fan-shaped OAB is expanded into a plane figure (1) What is the relationship between the length of the sector arc AB and the length of the circle at the bottom of the cone? What is the position relationship between point a and point B on the side of the cone? (2) If the angle ∠ AOB = 90 °, what is the relationship between the radius r of the cone bottom circle and the radius r of fan-shaped OAB (i.e. OA or OB)? (3) If point a moves on the side of the cone and then returns to its original position, how should the shortest distance of point a be designed? If R 2 = 0.5, ∠ AOB = 90 °, find the shortest distance of point a

(1) The arc length of the sector is equal to the perimeter of the bottom of the cone, and points a and B coincide on the side of the cone;
(2) ∵ the arc length of the cone is equal to the perimeter of the base,
∴2πr=90πR
One hundred and eighty
That is, r = 4R;
(3) If AB is connected, AB is the shortest distance;
∵r2=0.5
∴r=
One
2=
Two
Two
∵∠AOB=90°，
∴90πr2
360=πrR
The solution is: r = 2
Two
∵OA2+OB2=2R2=AB2，
∴AB=4
The shortest path length is 4

17. As shown in Fig. 24-a-10, the circular paper with radius of 2 is cut into two parts of 1:3 along the radius OA and ob, and the obtained sector is used to enclose the side of the cone Detailed explanation,

360/4=90 90*3=270 L=90π2/180=π L=270π2/=3π
Let the radius of small sector be r and that of large sector be r
Small sector = π = 2 π r large sector = 3 π = 2 π R
R=1/2 r=3/2

As shown in the figure, there is a fan with a center angle of 120 ° and a radius of 6cm. If OA and ob are overlapped to form a side of a cone, then the height of the cone is () A. 4 2cm B. 35cm C. 2 6cm D. 2 3cm

From the center angle of 120 ° and radius of 6cm,
The arc length of the sector is 2 π· 6
3=4πcm，
That is, the circumference of the bottom circle of the cone is 4 π cm,
The radius of the bottom circle is 2cm,
OA = 6cm,
According to Pythagorean theorem, the height of a cone is 4
2cm．
Therefore, a

As shown in the figure, the circular paper with radius of 2 is cut into two parts of 1:3 along the radius OA and ob, and the fan is used to enclose the side of the cone, then the bottom radius of the cone is () A. 1 Two B. 1 C. 1 or 3 D. 1 2 or 3 Two

As shown in the figure, there are two cases,
① Let S2 be the radius of the bottom of the cone as R2,
The central angle of S2 is 270 degrees,
Then its arc length is 270 π × 2
180=2πR2，R2=3
2；
② Let the radius of the bottom surface of the fan-shaped S1 made into a cone is R1,
The central angle of S1 is 90 degrees,
Then its arc length = 90 π × 2
180=2πR1，R1=1
2．
Therefore, D

As shown in the figure, the center angle of fan-shaped OAB is 90 ° and the diameter of OA and ob is taken as the semicircle in the sector. P and Q represent the two shadow parts respectively. Try to determine the relationship between the area of P and Q

∵ the center angle of sector OAB is 90 ° and the sector radius is a,
The fan area is 90 × π × A2
360=πa2
4，
The semicircle area is: 1
2×π×（a
2）2=πa2
8，
∴SQ+SM =SM+SP=πa2
8，
∴SQ=SP，
That is, the areas of P and Q are equal

As shown in the figure, the center angle of the fan-shaped OAB is 90 ° with OA and ob as the diameters, and the semicircle is made in the fan. P and Q represent the areas of the two shadow parts respectively. Then the size relationship between P and Q is () A. P=Q B. P＞Q C. P＜Q D. Can't be sure

∵ the center angle of sector OAB is 90 ° and the sector radius is a,
The fan area is 90 × π × A2
360=πa2
4，
The semicircle area is: 1
2×π×（a
2）2=πa2
8，
∴SQ+SM =SM+SP=πa2
8，
∴SQ=SP，
That is, P = Q,
Therefore, a

As shown in the figure, the center angle of the fan-shaped OAB is 90 ° and a semicircle is made in the sector with OA and ob as the diameter respectively. What is the relationship between the area of the two parts?

If the center angle of the sector OAB is 90 degrees, then
Sector area = π OA 2 / 4
If the diameter of OA and ob is taken as the diameter, the semicircle is made in the sector
Semicircle area = π OA 2 / 8
Area of two semicircles = π OA 2 / 4
Two semicircular areas = sector area
So the area of the overlapping part of two semicircles = the area of two semicircles and the sector without overlapping part

The circumference of a rectangle is 24 cm, and there are two circles of the same size inside. Calculate the area of the shadow part [outside the circle] To answer the process emergency

The length + width of a rectangle is 12 cm. From the sentence "there are two circles of the same size inside", we know that the width is 1 / 2 of the length
So the width is 4 cm and the length is 8 cm
4 * 8-4 / 2 * 4 / 2 * 3.14 * 2 = 32-12.56 * 2 = 32-25.12 = 6.88 square centimeter

As shown in the figure, O is the fulcrum. An object with a mass of 5kg is hung at the end of a, OA = 20cm, OB = 12cm, BC = 16cm, Ao is perpendicular to ob, ob is perpendicular to BC? And draw the schematic diagram of the force, and make the two arm of the lever

(1) Connecting OC is the longest power arm. According to the conditions of lever balance, the power direction of lever balance should be downward, and the minimum power can be drawn accordingly
②OC=
(OB)2+(BC)2=
(12cm)2+(16cm)2=20cm，
ν tension f = OA
OC×mg=20cm
20cm×5kg×9.8N=49N．
A: the force is 49n, and the drawing of the arm of force is shown in the figure above

It is known that: as shown in the figure, a point P, P1 and P2 in ∠ AOB are symmetric points about OA and ob respectively. P1p2 intersects OA with m and crosses ob with N. if p1p2 = 5cm, then the circumference of △ PMN is () A. 3cm B. 4cm C. 5cm D. 6cm

∵ P and P1 are symmetric about OA,
/ / OA is the vertical bisector of segment PP1,
∴MP=MP1，
Similarly, P and P2 are symmetric about ob,
/ / ob is the vertical bisector of segment PP2,
∴NP=NP2，
∴P1P2=P1M+MN+NP2=MP+MN+NP=5cm，
The circumference of △ PMN is 5cm
Therefore, C is selected