20 is what present of 30?

So the formula for a volume of a sphere is 4/3 times pi times the radius cubed.
step one: 4/3 times 3.14 = 4.18666666667
Step two: then you cube 8.5 and get 592.45
the final answer is 2,480.3906

6 0

3 years ago

Find integrate (1/(xsqrt(1+(lnx)^2)),1,e,x) ...?

prohojiy [21]

Note that x² + 2x + 3 = x² + x + 3 + x. So your integrand can be written as

(x² + x + 3 + x)/(x² + x + 3) = 1 + x/(x² + x + 3). 

Next, complete the square. 

x² + x + 3 = x² + x + 1/4 + 11/4 = (x + 1/2)² + (√(11)/2)² 

Also, for the x in the numerator 

x = x + 1/2 - 1/2. 

So 

(x² + 2x + 3)/(x² + x + 3) = 1 + (x + 1/2)/[(x + 1/2)² + (√(11)/2)²] - 1/2/[(x + 1/2)² + (√(11)/2)²]. 

Integrate term by term to get 

∫ (x² + 2x + 3)/(x² + x + 3) dx = x + (1/2) ln(x² + x + 3) - (1/√(11)) arctan(2(x + 1/2)/√(11)) + C 

b) Use the fact that ln(x) = 2 ln√(x). Then put u = √(x), du = 1/[2√(x)] dx. 

∫ ln(x)/√(x) dx = 4 ∫ ln u du = 4 u ln(u) - u + C = 4√(x) ln√(x) - √(x) + C 

= 2 √(x) ln(x) - √(x) + C. 

c) There are different approaches to this. One is to multiply and divide by e^x, then use u = e^x. 

∫ 1/(e^(-x) + e^x) dx = ∫ e^x/(1 + e^(2x)) dx = ∫ du/(1 + u²) = arctan(u) + C 

= arctan(e^x) + C.

4 0

3 years ago

Extension question (provide a full explanation of your method(s):

Volgvan

Answer:

Ann has little chance to win if she is presented with 4 counters.

Ann can always win from a pile of 6 counters.

(both are explained below)

Step-by-step explanation:

If Ann is presented with 4 counters, and

1. if she takes out 3, she will lose since the opponent will pull out 1 and the last one.

2. if she takes 2 her opponent will take out 1 and she can't pull out the last 1 since her opponents last move was to pull out 1 counter so she will lose.

3. If she takes out 1 and her opponent takes out 3 in the next move she loses.

but if instead of 3 her opponent takes out 2 and in the last move Ann takes out the last 1 then she will win.

So, If Ann is presented with 4 counters she has little chance to win provided in the move just before, her opponent didn't move 1 counter.

Now,

if there is 6 counters to Ann, and

1., if Ben's previous move was 1 then Ann can win if she takes out 3 or 2.

If she takes out 3 Ben can take out 1 or 2 and in the last move she will take out 2 or 1 (respectively) and winning the game.

If she takes out 2 Ben can take out 1 or 3 and in the last move Ann wins by pulling out 3 or 1 respectively.

2. if Ben's previous move was 2 then Ann can win if she takes out 1 or 3.

If she takes out 1 Ben can take out 2 or 3 and in the last move she will take out 3 or 2(respectively) and winning the game.

If she takes out 3 Ben can take out 1 or 2 and in the last move Ann wins by pulling out 2 or 1 respectively.

2. if Ben's previous move was 3 then Ann can win if she takes out 1 or 2.

If she takes out 1 Ben can take out 2 or 3 and in the last move she will take out 3 or 2(respectively) and winning the game.

If she takes out 2 Ben can take out 1 or 3 and in the last move Ann wins by pulling out 3 or 1 respectively.

7 0

3 years ago