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1/x + 1/y  =  1/18

Let  z  =  18

And   x, y  must be >  18

So  

Let x  =  z + a         and Let y  =  z + b

So we have that

  1              1              1

____  +   ____  =   _____     simplify

z + a       z + b           z

[ z + b + z + a]            1

____________ =   _______

[ z + a] [z + b]              z

[2z + a + b]                   1

_____________  =   _____               cross-multiply

 [ z+ a ] [ z + b]            z 

z [ 2z + a + b ]   =  [ z + a] [ z + b ]

2z^2 +az + bz  =    [ z + a ] [ z + b ]

2z^2 + az + bz   =  z^2  + az + bz + ab

z^2  =  ab

(18)^2  =  ab

324  =  ab

The factors  of  324   are  

1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 27 | 36 | 54 | 81 | 108 | 162 | 324 

The smallest possible value for a,b    is   12 , 27

So

x =  z + a  =   18 + 12  = 30

y =  z + b  =   18 + 27  = 45

So....the minimum value for x + y  =  75

It is a fraction button EP..

It does make fractions much easier to deal with.

But it is certainly not essential for anyone who has a good grasp of primary school maths.

It is not there because the designer of the calculator considered it unnecessary.

We have that

(x - 4) ( 2y - 1)   =   xy + 1       simplify

2xy - 8y - x  + 4  =  xy + 1

xy - 8y - x + 3   =   0

xy  =  x + 8y - 3

xy - x  =  8y -  3

x(y - 1)  =  8y - 3

             8y - 3

x =        ______

              y -  1

Possibilities  for  x, y

x           y

13        2              reject

9          6              accept

So  the product of your  numbers  =  9*6  =  54

If 14% are defective, then 86% are not defective.

Therefore, (0.86)⁷ is the probability that none are defective:  (0.86)⁷​  =  0.3479

At least one of them is defective is  1 - (none of them is defective):     1 - 0.3479  =  0.6521

The probability that at least one of them is defective  =  0.6521.

If both x and y must be whole numbers, then:  (x - 4)(2y - 1)  =  xy + 1

Multiplying out     --->        2xy - x - 8y + 4  =  xy + 1

Rearranging terms     --->     2xy - xy - 8y  =  x - 3

                                                      xy - 8y  = x - 3

Factoring     --->                            y(x - 8)  =  x - 3

                                                               y  =  (x - 3)/(x - 8)

If you test the number between 1 and 10 for x, you can get these whole number answers:  (3,0), (7,-4), and (9,6).

Of these, only the answer  x = 9  and  y = 6  have answers between 1 and 10 for both x and y.

See the link for my answer, GM

Here's my best effort, GM

(a)    x_i  =   1 + n*Δx         where  Δx  =   [ 5 - 1 ] / 4  =  1  

So

x_i = 1 + n(1)  =  1 + n

(b)  f(x_i )  =   ( 1 + n)^2 - 3

( c)   and (d)  = Right Endpoint Estimate

4                               4       

∑     f(x_i)  Δx   =        ∑       (1 + n)^2 - 3      =  42  units^2    

n = 1                        n = 1

(e)   Left Endpoint Estimate

3                               3       

∑     f(x_i)  Δx   =        ∑       (1 + n)^2 - 3      =  18  units^2   

n = 0                        n = 0

Using Calculus, the true area  is  29 + 1/3  units^2

(f) L₄  is an underestimate

(g) R₄  is an overestimate

Thanks, I was actually able to solve it right before you lol!

Didn't properly understand the question so I didn't draw a proper diagram; once I did the solution was obvious.

7 is to 8     as   x is to 10

7/8 = x/10

x = 7/8 * 10= 70/8 = 8 3/4 cm

If by similar you mean the area is the same then you find the area of the first.
10*8=80cm^2.

The second triangle then is 7*x=80 which can be rewritten as  x=80/7 => x=11.4.

 

because you can just do it easier by not using a button like that.

90 = 45 + 45 = ( 45 + 2.5 ) + ( 45 - 2.5 ) = 47.5 + 42.5 = 90  

What exactly does an  a(b/c) button do?     a*b/c  ?

Man, that was so easy to spot, I don't know why I didn't know that. Thanks CPhill!

We have that

x + ( x - 5)  =  90

2x - 5  =  90            add 5 to both sides

2x  =  95    divide both sides by 2

x =  47/5°

And the smaller  angle  is    47.5 - 5  =  42.5°

Assuming no restrictions (we can have empty boxes)....we have

n^k   ways     where n is the number of boxes and k is the number of balls....so...

4⁴  =  256 ways

thanks 

If the discriminant  =  0.....we have one  solution

If the discriminant > 0,  then we have two solutions

If the discriminant  is < 0, we have two non-real solutions

Let B   = L

Let A = L - 11

Let C  = (1/2)B    = (1/2)L  = .5L

So we have that

A + B + C   =  29

(L -11) + L + (.5)L  = 29         simplify

2.5L -  11  = 29             add 11 to both sides

2.5L =  40                     divide both sides by 2.5

L  = 16

A =  L - 11  =   16  - 11   =    5 cm

At least 8 means that   either 8, 9 or 10  heads were tossed

So  we have

C(10, 8)  + C(10,9)  + C(10,10)  =

45   +  10   +     1   =

56 possible sequences

Second one....

Yep.....  r   = 1/7    and it converges  

Your answer to the first one is good    !!!

Thanks ElectricPavlov!

Amount in container a at time t   is  800-6t

   amount in container b            is 1000-10t    when are they equal?

800-6t = 1000-10t

4t = 200

t = 50 minutes

Distance from center to corner (vertice) = sqrt (5^2 + 5^2 + 5^2) = sqrt 75

distance from center to surface is 5

sqrt75 - 5 = 3.66 units

A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?