All Answers 358083 Answers

You have listed them by "ascending" order, not by "descending" order !!!.

Alan is right !!!.
Compute the definite integral:

1134000000 integral_2^7.5 2^(-x/3) dx
For the integrand 2^(-x/3), substitute u = -x/3 and du = -1/3 dx.
This gives a new lower bound u = -2/3 = -2/3 and upper bound u = -7.5/3 = -2.5:
 = -3402000000 integral_(-2/3)^(-2.5) 2^u du

Switch the order of the integration bounds of 2^u so that the upper bound is larger. Multiply the integrand by -1:
 = 3402000000 integral_(-2.5)^(-2/3) 2^u du

Apply the fundamental theorem of calculus.
The antiderivative of 2^u is 2^u/(log(2)):
 = (26578125 2^(u + 7))/(log(2)) right bracketing bar _(-2.5)^(-2/3)

Evaluate the antiderivative at the limits and subtract.
 (26578125 2^(u + 7))/(log(2)) right bracketing bar _(-2.5)^(-2/3) = (26578125 2^(7 - 2/3))/(log(2)) - (26578125 2^(7 - 2.5))/(log(2)) = 2.22425×10^9:

Answer: | = 2.22425×10^9 Joules.

Your integral is incorrect.  It is not \(\int_2^{7.5}3^{-x}dx\)  It should be  \(\int_2^{7.5}(\frac{1}{2})^{x/3}dx\)

Also, since you previously used x for distance, it would have been clearer to use t in this integral.

(You and I obviously have a different concept of "messy"!!)

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I think your equation should look like this:

Let the number of jumps between Kola and Bola = n

1,533 - 1 =1,532 inches to be covered by 2-inch n jumps.

2n = 1,532 divide both sides by 2

n = 1,532 / 2

n = 766 number of 2-inch jumps between Kola and Bola, or 766 / 2 = 383 a piece.

766 + 1 = 767 total number of jumps between Kola and Bola, including the first 1-inch jump.

Jeff: Please do not be arrogant in trying to be a "show-off". As Melody says, you should be polite and thankful towards your elders. 

Not to be offensive or anything Jeffes but I think your response to Alan's answer is extremely offensive!

Whatever happened to "Thanks you very much, I really appreciate all you time and expertise"!!

Thanks Alan, that is a personal thank you from me :D

Mr. CPhill, can you please explain me in a details? I could not understand after 

"With the first three jumps, they have closed a distance of  1 + 2 + 2  = 5 inches So...there are 1533 - 5  = 1528 inches left between them". Please write in details. Plz.

Input:-1.5,2,-3,1/3 from least to greatest

Intepretation: List -1.5,2,-3, and 1/3 by ascending order:

By the rule that negative numbers are smaller than numbers: (Note that the higher the negative number is, the smaller it actually is.)

Answer:\(-3,-1.5,1/3,2\)

l x - 2 l ≤ 5.6

We have two equations here

x - 2  ≤ 5.6            and                x  - 2  ≥   - 5.6

add  2 to both sides

x ≤ 7.6                                     x ≥  -3.6

So.....the solution set is

-3.6 ≤ x ≤ 7.6

And there are 11 integers between these two values

I assume you want   1 / (16/9)

Note that we can actually wirte this as   (1/1)  / (16/9)

Since we're dividing fractions......invert the fraction in the denominator and multiply by the fraction in the numerator :

(1/1) * (9/16)  =    9 / 16

213412513452345

Corrections:  (i) I should have set t₁ to 2 years not 2/3 years!  The result is then E = 2.224*10⁹ J

                     (ii) The expression for E (after integration) should have t₁ and t₂ interchanged (or the result will be a negative energy!).

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Jeff, be sure to send Alan his $100.LOOOOL (Bitcoin should should work just fine).

Try not to confuse power (Watts), power density (Watts per square meter) and Energy (Joules)!

Here's my interpretation:

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Thanks, hectictar......!!!!

Another thing to realize is that circles are to one another as the squares of their radiuses.....

Thus.....the smaller circle has a radius of 1/2 the larger circle...so...its area  = (1/2)^2  = 1/4  that of the larger circle

Here's another method....

The sides are all equal, so one side  = 148/ 4  = 37

And the diagonals meet at right angles......so  1/2 the length of the known diagonal = 1/2 * 24  = 12

And this half diagonal  and one of the rhombus sides will form the leg and hypotenuse of a right triangle.....and the other half diagonal will form the other leg.....and its length  =  sqrt [ 37^2 - 12^2]  = sqrt [ 1225 ] =  35

So......the length of the other diagonal  = 2 * 35  = 70

Trouble factoring a quadratic?  Easy when you realise that   ----

 The factors of the constant term add up to the value of the co-efficient of  x

Here you have x^2 - 4x - 32.     We want the factors of   -32    that add up to      -4   because that is the

co-efficient  of x  here.

factors of -32  are plus or minus16 and plus or minus 2

and plus or minus 8 and plus or minus 4.         Since there is no way we can get - 4 from the sums of

plus or minus 16 and 2    we look at plus or minus 8 and 4.     The only possible sum is -8  + 4.

So the factors are  (x-8)(x+4)     Try a few more and you'll soon get the technique.

Mmmmm....and all along I thought "Q.E.D."   referred to  "Quit Eating Dinner !!! "

Here is the visual image of what you have described...

The question becomes....what "shaded area"  are you referring to ???

The sum of the digits of a two digit number is 9. the fractions formed by taking 9 less than the no.. as numerator and 9 more than the number as denominator is 3/4. what is the number

Let the digits be  a and b

Note that we can write any two-digit number as   10a + b

So we have this system

a  + b  = 9             

[ (10a + b) - 9]  / [ (10a + b) + 9]  = 3/4

Manipulating the first equation, we have that   b = 9 - a

Subbing this into the second equation, we have that

[ (10a + 9 - a) - 9] / [ (10a + 9 - a) + 9 ] = 3/4     simplify

[ 9a] / [ 9a + 18]  = 3/4        cross-multiply

4[9a] = 3 [9a + 18 ]   simplify

36a  = 27a + 54      subtract 27a  from both sides

9a  = 54     divide both sides by 9

6  = a

And b = 9 - a  = 9 - 6  = 3

So....our number is 63

Proof that this is correct :

[63 - 9] / [63 + 9 ] =    54 / 72   =  [3 * 18 ] / [ 4 * 18 ]  =   3 / 4

Let the number = a, then:

(a-9) / (a+9)=3/4

Solve for a:
(a - 9)/(a + 9) = 3/4

Cross multiply:
4 (a - 9) = 3 (a + 9)

Expand out terms of the left hand side:
4 a - 36 = 3 (a + 9)

Expand out terms of the right hand side:
4 a - 36 = 3 a + 27

Subtract 3 a - 36 from both sides:
Answer: | a = 63

Factoring and tell whether the parabola opens  up or down. Also Identify the minimum and how does the axis of symmetry  relate to the x-intercepts? 

y=x^2-4x-32

Note that the form  is y = x^2 + bx + c

factoring and tell whether the parabola opens  up or down. Also Identify the minimum and how does the axis of symmetry  relate to the x-intercepts? 

Since x^2 is positive and there  is no negative in front of the y, term, this parabola opens upward

Factoring  we have....  y =  ( x - 8) ( x + 4)

Setting   y  = 0, we can find the  x intercepts  thusly :   0 = (x - 8) ( x + 4).....setting each of these factors to 0 and solving for x  produces the x intercepts of x = 8   and x  = -4

The minimum  can be found thusly :

The x coordinate of the vertex  is given  by      -b / [ 2a ]   ....b = -4   and a = 1   .....so   ....   -b / [2a ] =  -[ -4] / [ 2(1)]   =

2

Now....we can find the y coordinate of the vertex  by plugging this value into the function...so we have ...

y = (2)^2 - 4(2) - 32  =  4 - 8 - 32  =  4 - 40  = -36    .....and this is the minimum y value of the parabola

The axis of symmetry will be found between the  intercepts  and is given  by adding the intercepts and dividing by  2....so we have ... [ 8 + -4 ] / 2 =  4 / 2  = 2.......so the axis of symmetry   is x = 2....this is no coincidence.....the axis of symmetry will occur at the vertex which is ( 2 , - 36)

Kola and Bola are two friends. They are staying on two sides of a pond. Now Kola jumped 1 inch first to reach Bola. Seeing that, Bola jumped double distance of Kola. This time, Kola jumped equal distance of Bola. Thus they could meet each other by jumping equal distance. If the distance of two sides of the pond is 1533 inches, then how many jumps they have given totally?

With the first three jumps, they have closed a distance of  1 + 2 + 2  = 5 inches

So...there are 1533 - 5  = 1528 inches left between them....if each jumps 2 inches at a time.....they will each jump a total of

1528 / 2  = 764 inches......so....it will take each   764 / 2 = 382 jumps to meet each other  [  assuming that they can jump in water !!! ]