Mathematics, extra points for getting it right within 20 minutes.

1.Proof: Let ab be any diameter in the garden O, and Cd is any string in the circle, and AB must pass through the center of the circle O by the definition of diameter, and connect OC and OD, then in the triangle OCD, the sum of any two sides of the triangle is greater than the third side, OC+OD is greater than CD, and OC=OD=OA=ob=1 2AB, so AB is greater than CD.

That is, the diameter is the longest string in the circle.

cos∠oab=18/30=3/5

Assuming that the diameter ab is not the longest string in o, there must be a chord cd>ab

Linking co, do, then co+do=ab, co+do>cd, (the sum of the two sides in the triangle is greater than the third side) ab>cd

This contradicts the assumption that CD>AB is the longest string in O.

1) Do od ab

With the sag diameter theorem, AD=1 2AB=1 2 36=18 (mm) in the RT triangle AOD.

From the Pythagorean theorem, you can find od (do the math yourself).

2) The cosine value of the angle OAB is 18 30

1. Question 1 Counterproof Assuming that there is a string with a length greater than the diameter, which connects the center of the circle and the end of the chord respectively, then a triangle is formed, and the sum of the two sides of the triangle is equal to the third side, so the assumption is wrong and proved.

Do OC perpendicular to AB to C, C bisects AB, OC = root number (30 2-18 2) = 24mm

cos angle oab = 18 30 =

1.A string can be made at any end of any diameter in a circle to form a right-angled triangle. I've heard that the hypotenuse is larger than the right-angled edge. Certification.

2 (1) Extend AO to intersect the circle at M, connect BM, pass O to make AB perpendicular line, cross AB at N, you can know that the angle ABM is a right angle, and the angle ANO is also a right angle, the length of ON is used, using the similarity relationship, on=1 2BM, using the Pythagorean theorem we can know that MB=48mm, so the on=24mm sought

2）cosoab=ab/am=3/5

Base radius = meters.

Volume = 3 * 3 * cubic meters.

The first type: do the bottom circumference.

Then radius r=m.

Volume = cubic meters.

The second type: rice to make the circumference of the base.

Then radius r=m.

Volume = cubic meters.

So the maximum volume is cubic meters.

It is a meter high and enclosed in a column.

Radius = cylinder volume = 3 squared * cubic meter.

Since the area of the circle is r squared), the larger the radius, the larger the volume of the cylinder.

It's very simple, calculate the circumference of the long ground, and calculate the circumference of the wide ground to know,

The meaning of the question is that the supermarket is the shortest distance from point A, B, C, D, assuming that the supermarket is the shortest distance of point e = ae + be + ce + de

Combining the plots, we can see that E gets the shortest distance between the CD segments.

So the best solution is to build the supermarket (E) between CDs.

The supermarket is built at the midpoint of AB.

If ab is 9, then ac is 3, cd is 3, db is 3, and if it is built at the midpoint of ab, the distance to a and b is and the distance to c and d is so that every person in the neighborhood is closest to the supermarket.

If the supermarket is to be built on section AB, the distance that residents of the two places AB must walk together is exactly the distance of section AB, and it is obvious that it must be built on the CD, and it is appropriate to build it in the middle of the CD for the sake of fairness.

S A = S shadow 48%.

s B = s shadow 72%.

s C = s shadow 56%.

1) The area of A is a few percent of the area of B.

s A s B = (s shadow 48%) s shadow 72%) = 72% 48% = 3 2=

2) The area of C is a few percent of the area of A.

S C S A = (S Shadow 56%) S Shadow 48%) = 48% 56% = 6 7

S A = S shadow 48%.

s B = s shadow 72%.

s C = s shadow 56%.

If the area of the common part is 1, then the area of A is 1 48%, the area of B is 1 72%, and the area of C is 1 56%, A: The area of A is the area of B, and the area of C is the area of A.

If the area of the common part is 1, then the area of A is 1 48%, the area of B is 1 72%, and the area of C is 1 56%, so the area of A and B = (1 48%) (1 72%) = 72 48=

Area of C Area of A = (1 56%) (1 48%) = 48 56=

Solution: Let the total area of A be x, the total area of B be y, and the total area of C be z.

Then: 48%x=72%y=56%z

1) x÷y=72%÷48%=

2）z÷x=48%÷56%≈

Answer: The area of A is the area of B, and the area of C is the area of A.

Let the area of A be x, the area of B is y, the area of C is z, and then 48%x=72%y=56%, and then just find it, I won't count it for you. . . I hope you're doing your best in math...

Think of the common area as 1

The area of A is 1 48% = 25 12

The area of B is 1 72% = 25 18

The area of C is 1 56% = 25 14

1) (25 12) (25 18) = 3 2 = 150% (2) (25 14) (25 12) = 7 6 = (approximately equal to).

Solution: s A = s shadow 48%.

s B = s shadow 72%.

s C = s shadow 56%.

1) The area of A is the area of B.

s A s B = (s shade 48%) (s shade 72%) = 72% 48% = 3 2 = 150%.

2) The area of C is the area of A.

S C S A = (S Shadow 56%) S Shadow 48%) = 48% 56% = 6 7

1.After cutting, the cross-section is a square with a side length of 4 decimeters, and it can be seen that its diameter is 4 decimeters, and its height is also 4 decimeters.

Two base areas = 2* *r 2=2* *4 2) 2=8 square decimeters.

Side = circumference of the round bottom * height = *4 * 4 = 16 square decimeters.

Therefore, the original cylindrical wood surface area = 8 + 16 = 24 square decimeters.

2.If the side length is a, the height is a, and the circumference of the base area is also.

Therefore, the diameter of the bottom area = a.

Base area diameter: height = a : a = 1 .

1.Because the cross-section is a square with a side length of 4 decimeters, the diameter of the bottom surface is 4 decimeters, and the height is 4 decimeters.

2.Because the back is square, the bottom circumference = height, and because the bottom diameter = perimeter

So simplify it for 1 to 1 and come up with the answer.

1. The height of the cylinder is 4

Base diameter = 4, radius = 2

Surface area = square decimeters.

2. Circumference = height.

r=hr：h=1：π=:157

1. The cross-section is a square with a side length of 4 decimeters, which means that the side length of the square is 4 4 = 1 decimeter, that is, the height of the cylinder is 1 decimeter, the diameter of the bottom surface of the cylinder is 1 decimeter, and the circumference of the bottom surface of the cylinder is 1 = , then the side area of the cylinder is 1= A base area of the cylinder is (1 2 1 2) = 1 4 The surface area of the cylinder is the side plus two bottom surfaces: s = +1 4 = 5 4 (square decimeters).

2. The side area of the cylindrical shape happens to be a square, which means that the circumference and height of the bottom surface of the cylindrical shape are equal, and the diameter of the bottom surface is r, and the circumference of the bottom surface is r, then the height of the cylindrical shape is also r

The ratio of the diameter of the base surface to the height of the cylinder is: r ( r) = 1 = 1

1200-397 = 1200 - 400 +3 = 803578+298 = 578 + 300 -2 =87616÷ = 16×4 ÷（

= (not short).

3/3 + 5/5 = (+5 8) = 12 - 1 = 11 = 4 (125% 8) = 10 10 = 100

1200-397

16÷=（16x4)÷（

3 out of 3 + 5 out of 5

3/3 + 5/8).

10x10

1200-397

3 out of 3 + 5 out of 5

3/3 + 5/8).

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