1. Let the upper base length of the trapezoid be a, the lower base length be b and the height be h, then the trapezoid area is (a+b)h/2. According to the meaning of the question, the following equations can be listed:

A+3 = b-3 (upper sole 3cm, lower sole 3cm)

B-a = 8 (become a square)

(a+b)h/2 = S (trapezoidal area)

Solve the equations and get a= 13, b=2 1, h=6, S= 1 14.

So the area of this trapezoid is 1 14 cm2.

2. Let the distance between east and west be x, then car A walks for y hours and car B walks for z hours, then the following equation can be listed:

54y+46z = x (the place where the two cars meet is the sum of the distances of Y hours in the east and Z hours in the west).

Y+z = x/28 (when two cars meet, the distance from the midpoint to the two places is 1/2 of the total distance).

Y = 28-z (substituted in the above formula)

Solve the equations and get x=2440.

So the distance between east and west is 2440 meters.

3. Let a rectangle have a length of L and a width of W ... According to the meaning of the question, the following equations can be listed:

2l + 2w = 22

L 2+W 2 = H 2 (the hypotenuse of a triangle represents the diagonal of a rectangle)

Rewrite the first equation as l = 1 1-w, and substitute it into the second equation to get:

( 1 1 - w)^2 + w^2 = h^2

2w^2 - 22w + 12 1 = h^2

Because you need to draw the largest triangle, you need the maximum value of h. Note that h is related to w, so it is necessary to derive the second equation:

dh/dw = 4w - 22

Let dh/dw = 0 to get w = 5.5, and substitute it into the second equation to get h = sqrt(60.5).

So the largest triangle area is 1/2lw = 24.75 square centimeters.

4. X 5m long steel pipes and Y 3m long steel pipes are needed. According to the meaning of the question, the following equations can be listed:

5x+3y = 4 1 (total length is equal to 4 1m)

Y = 4 1-5x (use 5m long steel pipe as far as possible).

Substituting the second equation into the first equation gives:

5x + 3(4 1 - 5x) = 4 1

2x = 8

x = 4

Therefore, four 5m long steel pipes and five 3m long steel pipes should be used.

5. If the number of students attending the tea party is X, the following equation can be listed:

x/2 + x/3 + x/4 = 65

Divide by the denominator and move the term to get:

23x/ 12 = 65

x = 3 12/23

Therefore, about 13 students attended the tea party. In fact, because the number of students required by the topic is an integer, it should be rounded up, that is, the number of people attending the tea party is 14.