A mathematical problem, to calculate the formula Uncle Lin made a cylindrical lantern with a diameter of 20cm at the bottom and a height of 30cm. There are 78.5cm openings between the upper and lower floors. How much colored paper did he use

A mathematical problem, to calculate the formula Uncle Lin made a cylindrical lantern with a diameter of 20cm at the bottom and a height of 30cm. There are 78.5cm openings between the upper and lower floors. How much colored paper did he use

Bottom radius = 20 △ 2 = 10 (CM)
Bottom area = 10 & # 178; × 3.14 = 314 (cm2)
Side area = 20 × 3.14 × 30 = 1884 (cm2)
Required paper: bottom area + side area - reserved area
=2×314+1884-2×78.5
=628+1884-157
=2355 (square centimeter)

If 0 < α < π, then (1 + sin α + cos α) (sin α 2 − cos α 2) (2 + 2cos α)=______ ．

∵ 0 ＜ α ＜ π, ∵ 0 ＜ α 2 ＜ π 2, ∵ cos α 2 ＞ 0, the original formula = (2Sin α 2cos α 2 + 2cos 2 α 2) (sin α 2 − cos α 2) 2 × 2cos 2 α 2 = 2cos α 2 (sin α 2 + cos α 2) (sin α 2 − cos α 2) 2cos α 2 = sin2 α 2-cos 2 α 2 = - cos α

Simplify a formula
Double root 2-1 + 2 / 2 root

Double root 2-1 + 2 / 2 root
=2 radical 2-1 + 2 radical 2 / 2
=2 radical 2-1 + radical 2
=3 root 2-1

Help me simplify this formula
[(x^2+3x+9)/(x^2-27)]+[6x/(9x-x^2)]-[(x-1)/(6+2x)]
（√18-4√1/2+1/√2-√3）/(√3/3)

=[(x+3)^2/(x^2-27)+6x/x(3-x)(3+x)]-[(x-1)/2(3+x)]
Recanalization is simple
The feeling (x ^ 2-27) should be x ^ 2-9

2a（1＋x%）²-a（1-x%）²

2a（1＋x%）²-a（1-x%）²
=a*[2(1+2*x%+ x%*x%)- (1- 2*x%+x%*x%)]
=a*[2+4*x%+ 2(x%)²- 1 + 2*x%-(x%)²]
=a*[1+6*x%+ (x%)²]

To help simplify the second radical of a question: - root 7 ／ 3 root 15 / 14 × 2 / 3 root 2 and 2 / 1

-√ 7 ／ 3 √ (14 / 15) × (3 / 2) √ (2 and 1 / 2) = - √ 7 ／ 3 √ (14 × 15 / 15 & # 178;) × (3 / 2) √ (5 / 2) = - √ 7 ／ 3 √ 210 / 15 × (3 / 2) √ (10 / 2 & # 178;) = - √ 7 ／ 210 / 5 × (3 / 2) = -

Given the function f (x) = a ^ x + (X-2) / (x + 1), (a > 1), it is proved that the function f (x) is an increasing function on (- 1, + ∞)

Ling-1

Simplify a radical
How to simplify 1 / ((3) ^ (1 / 3)), that is, how to rationalize the denominator,

=3^(2/3)/[3^(1/3)×3^(2/3)]
=3^(2/3)/3^(1/3+2/3)
=3^(2/3)/3

Who can help me simplify the sequence summation
An = (2n-1) x (the N-1 power of 1 / 2) I know the method is not right

An=(2n-1)*(1/2)^(n-1)=2*n*(1/2)^(n-1)-(1/2)^(n-1);
Let BN = 2n * (1 / 2) ^ (n-1); CN = (1 / 2) ^ (n-1); then an = BN CN; San = SBN SCN;
Sbn=B1+B2+B3+B4+.+Bn
1 / 2sbn = 1 / 2B1 + 1 / 2B2 + 1 / 2b3 +. + 1 / 2bn (dislocation subtraction, bn-1 / 2B (n-1) = 2 * (1 / 2) ^ (n-1) = 4 * (1 / 2) ^ n);
1/2Sbn=Sbn-1/2Sbn = 2+{4*(1/2)^2+4*(1/2)^3+.+4*(1/2)^n}-2n*(1/2)^n
=2+(1-(1/2)^(n-1))/(1-1/2)-2n*(1/2)^n=2+2*(1-(1/2)^(n-1))-2n*(1/2)^n
Sbn=8+(1/2)^(n-3)-n*(1/2)^(n-2);
Scn=[1-(1/2)^n]/(1-1/2)=2-(1/2)^(n-1);
San=Sbn-Scn=8+(1/2)^(n-3)-n*(1/2)^(n-2)-(2-(1/2)^(n-1))
=6+(1/2)^(n-1)+(1/2)^(n-3)-n*(1/2)^(n-2)

(1-x) + (1-y) = (x + y)
Why is it equal to this number? How can I get it
This is the original formula, which is reduced to y = 1 + 1 / X-X
Why do you get y = 1 + 1 / x-x

1-X）²+（1-Y）²=（X+Y）²
1-2X+X²+1-2Y+Y²=X²+Y²+2XY
1-X-Y=XY
Y=（1-X）/(1+X)
That's it