All Answers 358090 Answers

First one

P(X≤2)  = (1/2)(2) (0.2)  = 0.2

P(X≥4)  =  (1/2)(1)(0.4)  = 0.2

P(2≤X≤4)  =  1 - P(X≤2) - P(X ≥ 4)  =  1 - 0.2 - 0.2  =  0.6

P(1≤X≤3)  = P(X≤3)  - P(X ≤1) = (1/2)(3)(0.3) - (1/2) (1) (0.1) =  0.4

Second one

a = 4

To see why......draw a vertical line at X = 4

This will form two trapezoids  with bases   4  and 1  and height .20

So....the area  of one of these trapezoids  =  (1/2)(.20) (4 + 1)  =  .10 * 5  =  0.5

So  P(X ≤ 4)  = .50

Your layout is appauling AND there is no actual question.

How many base-10 integers are exactly 4 digits in their base-3 representation and exactly 2 digits in their base-6 representation?

4 digits in base 3       

1000 base 3 = 3^3 = 27 base 10

2222 base 3 = 2*27+2*9+2*3+2 = 80

So it has to be between 27 and 80 base 10

and exactly 2 digits in their base-6

10base 6 = 6 base 10

55base 6 = 6^2-1 = 35 base 10

So it has to be between 6 and 35 inclusive base 10

So to meet both these requirements the number must be between  27 and 35 base 10 inclusive.

So 9 numbers qualify.

Note that    x + y           can be written as      1  +  y

                   ____                                                  ___

                      x                                                       x

So....we need to minimize        y     ......  and  we need the smallest possible negative

                                                 __

                                                  x

This will occur when y  =  2  and x  = -4

So....the largest possible value  for          x + y           2 + -4           -2             1

                                                                _____  =    _______ =    ___   =      __

                                                                    x                  -4             -4              2

7x^3 + 2y^3 - 3y +3 is the answer.

~~Hypotenuisance

2y^3 + 4y^2 - 3y - 1 is the answer.

~~Hypotenuisance

Note that if we draw a vertical line at x = 3.....we have a triangle on the right with a base of 1 and a height of .25

So....the area of this triangle  =  (1/2) (base) (height)  = (1/2) (1) (.25)  = .125

So....P ( X < 3)  =   Total area -  area of the triangle on the right  =  1 - .125  = .875

sin (30°)  =  1/2

But....the cosine (120°)  =   -cos(60°)  =  -1/2

The cosines of (1), (2) and (3)    = 1/2

So......(4)  is correct

GM....the cosine cannot  be  4  / sqrt (11)

The max value for the cosine  = 1....but  4 /sqrt (11)   ≈  1.206.....and this isn't possible

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Yah!    See CPhill's comment below....

Hectictar, I have a question...

Isn't there 1024 MB in a GB??

~~Hypotenuisance

The answer to the problem is 6. There are 6 numbers, 8, 9, 10, -8, -9, and -10 that satisfy this requirement of this problem.

~~Hypotenuisance

Let a, b, and c be 1, 2, and 3 respectively. Substituting the expression woukd give you 1 + 4 + 9. That equals 14.

~~Hypotenuisance

In other words... what is being satisfied in the problem??

~~Hypotenuisance

What is the correct question you are asking???

~~Hypotenuisance

The answer to this question is 2x^2 or \(2x^2\).

~~Hypotenuisance

You need to add brackets to make the meaning clear.

Hi Max

That is an interesting answer but...

Where did you get 10+10+17 from?

 .

Ohhhhhhh, I never thought of it like that. That is a brilliant way of solving that!

Thanks