2. Let the side length x, the side L=2X, and the included angle a of the regular triangle at the bottom.

Height of regular triangle H=Xsin60 degrees =√3/2X.

Height of the outside of isosceles triangle h=√(L? -(x/2)？ )=√[(2x)？ -(x/2)？ ]=√(4x？ -x？ /4)=√ 15x/2

cosa=(H？ +L？ -h？ )/(2HL)=(3/4x？ +4x？ - 15/4x？ )/(2√3/2x * 2x)= 1/(2√3)=√3/6

3.f(x)= a b = 1 *( 1+sin2x+cos2x)+ 1 *( 1)= sin2x+cos2x = sin2x+sin(π/2-2x)

= 2 sinπ/4 cos(2x-π/4)=√2 cos(2x-π/4)

Domain: r, range [-√2, √2], minimum positive period =2π/2=π.

4.3Sn=5an-an- 1+3Sn- 1，3(Sn-Sn- 1)= 5an-an- 1，3an=5an-an- 1

2an = an- 1, an/an- 1 = 1/2 equal ratio series.

( 1)an=a 1q^(n- 1)=2*( 1/2)^(n- 1)=2^(2-n)

(2)b 1=2^ 1=2 bn=4/an=2^n

tn=b 1(q^n- 1)/(q- 1)=2(2^n- 1)/(2- 1)=2(2^n- 1)

5.2a=√2，a=√2/2

e=c/a=c/(√2/2)=√2/2，c= 1/2

Answer? -B? =c？ ， 1/2-b？ = 1/4，b？ = 1/4，b= 1/2

( 1) x？ /( 1/2)+y？ /( 1/4)= 1，2x？ +4y？ = 1

(2) Right focus (c, 0), namely (1/2, 0)

L:y=x+b，0= 1/2+b，b=- 1/2

y=x- 1/2

Substitute into elliptic equation: 2x? +4(x- 1/2)？ = 1，2x？ +(2x- 1)？ = 1

2x？ +4x？ -4x+ 1= 1，6x？ -4x=0，x 1=0，x2=2/3

y 1=- 1/2，y 1 = 2/3- 1/2 = 1/6

A(0，- 1/2)，B(2/3， 1/6)

AB=√[(0-2/3)？ +(- 1/2- 1/6)? ]=√[4/9+4/9]=2√2/3

6. Assuming an increase of X yuan, the sales volume will decrease by 10X, and the actual sales volume will be 500- 10X.

And the profit of each piece =(50-40)+X= 10+X yuan.

Total profit y = (10+x) (500-10x) =10 (10+x) (50-x) =10 (500+40x-x). )= 10(500+400-400+40X-X？ )

= 10(900-(20-X)？ )

When X=20, y max = 10*900=9000 yuan.