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Let each small cube be  1 cubic unit.

volume of big cube  =  125 cubic units

side length of big cube  =  125^(1/3) units  =  5  units

surface area of big cube  =  total painted area  =  6 * 5^2  =  150  sq units

surface area of one small cube  =  6  sq units

combined surface area of small cubes  =  125 * 6  sq units  =  750 sq units

Each face of small cube is 1 sq unit.

There are a total of 150 sq units that are painted, so there are a total of 150 faces with paint on them.

There are a total of 750 sq units, so there are a total of 750 faces.

The probability that a randomly selected face is one with paint on it  =  150/750  =  1/5

Correct! Thank you so much!

f(x)  =  7x + 5

g(x)  =  x - 1

h(x)  =  f( g(x) )  =  f( x - 1 )  =  7(x - 1) + 5  =  7x - 7 + 5  =  7x - 2

h(x)  =  7x - 2

                            To find the inverse, let's replace  h(x)  with  y .

y  =  7x - 2

                            Solve for  x  ....add  2  to both sides of the equation.

y + 2  =  7x

                            Divide both sides by  7 .

(y + 2)/7  =  x

x  =  (y + 2)/7      So the inverse of  h(x)  is..

h^-1(x)  =  (x + 2)/7

By the same process used in your question here:https://web2.0calc.com/questions/please-help_69323

Height =7.93725 inches.

OK, I will take a crack at it! Let me know if I s***w up!!

Sqrt(288) = 16.97 cm - side length of the square base 

By Pythagoras' theorem:

Slant Height =sqrt[15^2 - (16.97/2)^2]

Slant Height =12.3695 cm

Height =sqrt[12.3695^2 - 8.485^2]

Height =9 cm

Volume = 1/3 x H x Side Length^2

Volume = 1/3 x 9 x 16.97^2

Volume =~864 cm^3

I'm confident i've got the correct ODE. (part a)

$\frac{di}{dt} + \frac{1}{RC}i = \frac{d{V}_{s}}{dt}\frac{1}{R}$

I come unstuck with part b

I'm not sure how to solve it with a ${V}_{s}(t) = 12t$

If I use the non homogeneous methord to solve it,

I get a ${y}_{H} = i(t) ={C}_{1}{e}^{-\frac{1}{RC}t}$

and then I don't know what my guess is for the RHS = $\frac{d{V}_{s}}{dt}\frac{1}{R}$

have I got the incorrect ODE in the first place?

any help is greatly appreciated!!

If you wish to spend about 50% of your money on chicken and about 50% on steak, then you have:

$50 / $1.29 =~39 Lbs of chicken

$50 / $3.49 =~14 Lbs of steak.

So that:

[39 x $1.29] + [14 x $3.49] = $99.17 - total amount of money spent.

$100 - $99.17 = $0.83 - money left over.

Note: if you want to spend all of $100 on either chicken or steak, you would simply divide $100 by $1.29 =~77 Lbs of chicken. Or, $100 divided by $3.49 =~28 Lbs of steak. In either case, you will have:

$0.67 cents and $2.28 respectively left over. With $2.28 you could buy one extra pound of chicken for $1.29, which will leave you with $0.99 cents left over.

Hi, I can help you :)

The points A (6,1) and B(-2,5) are on the line with equation y=-1/2x+4 

M is the midpoint of AB

Find an equation of the line through M that is perpendicular to y=-1/2x+4

the gradient of the line   y=-1/2x+4   is    $-\frac{1}{2}$

The gradient of the line that is perpendicular is the negative reciprocal which is    $+\frac{2}{1}=2$

So 2 will be the gradient of the new line.

The midpoint of AB is simply  the average of the xes and the average of the ys     

$midpoint=(\frac{6+-2}{2},\frac{1+5}{2})\\ midpoint=(\frac{4}{2},\frac{6}{2})\\ midpoint=(2,3)\\$

So you want the equation of the line through (2,3) with a gradient of 2

there are lots of methods to do this.

Here is one, my start is a bit different from normal, I think it makes it easier because it is the same for many different questions. You just need 2 different ways to express the gradient to make it work.

$\begin{align} gradient &=gradient\\ m&=\frac{y-y_1}{x-x_1}\\ 2&=\frac{y-3}{x-2}\\ 2(x-2)&=y-3\\ 2x-4&=y-3\\ 2x-1&=y\\ y&=2x-1 \end{align}$

I hope all that helps :)

Let the number of Liters of 20% solution = T

12%*14 + 20%*T = 13%[14 + T]

1.68 + 0.20T = 1.82 + 0.13T      subtract 0.13T from both sides

1.68 + 0.07T = 1.82                   subtract 1.68 from both sides

0.07T = 0.14                              divide both sides by 0.07

T = 2 - Liters of 20% solution will be needed to make 13% solution.

Very, very nice, detailed solution, hectictar! Bravo!

What is the inverse?

A slight typo !! Should be = 100 / 101

Note  that 

1 + 3 + 5 + 7 +.......+ 199     has  100 terms

And the sum of these =  n^2  = 100^2  = 100 * 100

And  

2 + 4 + 6 + 8 + ......+ 200   also has  100 terms

And the sum of these  is  n(n + 1)   =  100 (101)

So

[  1 + 3 + 5 + 7 + ....+ 199 ]            100 * 100                  100

______________________  =  ___________   =          ___ 

[ 2 + 4 + 6 + 8 +......+ 200]             100 * 101                  101

These problems easily confuse me but here's my attempt....

surface area of N's donut holes   =   4 pi * (6 mm)²   =   144pi   mm²

surface area of T's donut holes   =   4 pi * (8 mm)²   =   256pi   mm²

surface area of A's donut holes   =   4 pi * (10 mm)²   =   400pi   mm²

When N finishes  1  donut, T will have finished  $\frac{144}{256}$  donuts and A will have finished  $\frac{144}{400}$  donuts.

When N finishes  1  donut, T will have finished  $\frac{9}{16}$  donuts and A will have finished  $\frac{9}{25}$  donuts.

When N finishes  2  donuts, T will have finished  2*$\frac{9}{16}$  donuts and A will have finished  2*$\frac{9}{25}$  donuts.

When N finishes  2  donuts, T will have finished  $\frac98$  donuts and A will have finished  $\frac{18}{25}$  donuts.

When N finishes  x  donuts, T will have finished  $\frac{9x}{16}$  donuts and A will have finished  $\frac{9x}{25}$  donuts.

We need the smallest integer  x  such that  $\frac{9x}{16}$  is an integer and  $\frac{9x}{25}$  is an integer.

For a fraction to be an integer, all the factors in the denominator must be in the numerator.

For  9x/16  to be an integer,  2*2*2*2  needs to be factors of  x  .

For  9x/25  to be an integer,  5*5  needs to be factors of  x .

So  x  must be  2*2*2*2*5*5   =   400

When Niraek finishes 400 donut holes, Theo will have finished  400*9/16 = 225  donut holes, and Akshaj will have finished  400*9/25 = 144  donut holes.

$f(x) = \frac{3x - 7}{x + 1}$

Write

y  =  [ 3x - 7]  /   [ x + 1]       get x by itself

y [ x + 1]  = 3x - 7

yx + y  =  3x - 7

y + 7  =  3x - yx

y+ 7  =  x (3 - y)

[y + 7 ] / [ 3 - y ]  =  x     "swap" x and y

[ x + 7 ] / [ 3 - x ]   = y   =  f^-1(x)

I'm using  "e" for  "x"

Let   a < b < c < d < e

So we have......

a + b + c + d + e  = 320   (1)   

Where "a" is the smallest integer and "e" is the largest

And

c + d + e  =  283    (2)  

a + e  = 119     (3)

Sub (2) into (1)

a + b + 283  =  320

a + b  =  37

Let  a  =  1

Then.....as large as "e" can be is

1 + e  = 119

e  = 118

Let  a  = 36

Then......  a possibility for as small as "e" can be  is

36 + e  = 119

e  = 83

However     c < d < e

And  c + d + e  = 283

So....e will be minimized when  the sum of c and d are as large as possible

If      c  = d = e = 95

95 * 3  =  285....but  c and d  are < e.....so....c and d < 95  and  c < d

So....the largest that the sum of c and d can be  is  93 + 94  =   187

But.....

93 + 94 + 95   =  282   ......too small

Then e cannot be < 96  because  93 + 94 + 96  = 283

So.....as  small as e  can be  is  96   and as large as e can be  is  118

So.....the positive difference   between  the greatest positive value of  e   and the least positive value of e  is   118 - 96   =  22

Multiplying the  x^4 * 2 x^4  results in  2x^8       It will be 8th degree

Danke Omi67! Oh, wusste gar nicht dass es auch eine Englische Seite gibt

Thaks Omi......here's another approach....

By the Law of Cosines, we have

9^2  = 7^2 + 8^2  -  (2 * 8 * 7)cos(DCB)

-32   / -112  =  cos(DCB)

2/7  = cos(DCB)      (1)

And since angle DAB is supplemental to DCB, cos DAB  = -cos(DCB)

So....by the Law of Cosines again, we have

9^2 =  7^2  + AB^2  - (2 * 7 * AB) (-cosDCB)

[32- AB^2] = (14AB) cos(DCB)

[32 - AB^2] / [14AB ]   = cos(DCB)   (2)

Set  (1)  = (2)

2/7  =  [32 - AB^2] / [ 14AB ]

2 = [32 - AB^2] / [ 2AB]

4AB = 32 - AB^2

AB^2 + 4AB - 32  =  0      factor

(AB + 8) (AB - 4)  = 0

Setting both factors to 0 and solving for AB produces  AB  = -8  or AB = 4

Reject the first answer, accept the second

Dropping a perpendicular line from B to intersect DC at E.....then, by symmetry,  EC  =  2 

The height of the trapezoid  is given by √[BC^2 - EC^2 ] =  √[7^2 - 2^2]  = √45 = 3√5

So....the area is

height / 2   *  (sum of the bases )  =

[ 3√5  ( 8 + 4) ]  / 2  =    18√5  units^2  ≈  40.25 units^2

Hallo ellie,

warum stellst Du deine Frage nicht auf der deutschen Seite?

Jetzt schicke ich dir erst mal die Ableitungen. Der Rest kommt später.

Alcumus is never due. And stop using this site to copy and paste questions from aops to get an answer.

Hey, what is the 1,2,3 derivative of: 1/4 (4-x) * e ^ x? And how do I determine arithmetically extrema, inflection points of f and intercept points of f and f '?

you have to definitely use v and v 'and u and u' somewhere but unfortunately I do not know exactly where

Thank you in advance    [From Google Translate]!.

There are: 12, 23, 34, 45, 56, 67, or 6 combinations where the difference between them is < 2

21 - 6 = 15 combinations where the difference between them is 2 or > 2

15 / 21 = 5/7 probability that the difference between 2 randomly drawn numbers will be 2 or greater.

The numbers  are written down and some of them are colored blue. A coloring is called factorific if there is at least one blue number, and for each blue number, all of its divisors are also blue.

If Grogg randomly colors some, all, or none of the numbers from 1 to 6 blue, what is the probability that his coloring is factorific?

I have made an assumtion here. The likelihood that any individual number is blue is 50%

For this to work number 1 must be blue because it is a factor of all the others.

Since 1is blue, it really doesn't matter whether 2,3 or 5 are blue as their only factor other then themselves is 1.

6 blue , 3blue, 2 blue, 1 blue others don't matter.   Prob = 0.5^4 = 1/16

6 not blue, 4 blue, 2 blue, 1 blue others don't matter  Prob = 1/16

6 not blue, 4 not blue, 1 blue and others don't matter  Prob = 1/8

1/16  +  1/16  +  1/8 = 4/16  = 1/4 

I think that the chance that his colouring is factoric is  1/4

pH = -log[H+] 

pH = -log[3.3x10^-3] 

pH = 2.5