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10 paintings fall outside the circles....so....50 fall inside...and we have

x(x - 2) + x + 2x + 8  = 50

x^2 -2x + 3x = 42

x^2 + x  - 42  =  0    factor

(x + 7) ( x - 6)   = 0

Setting  both factors to 0 and solving for x gives  x  = -7 (reject) or x  = 6

So...the total number of painitngs from the 20th century  is x * (x - 2) = 6* (6 - 2)  = 24   (1)

And of these,  x = 6  were British    (2)

So....the probability that a painting is British given that it is from the 20th cenury  is  (2) / (1)  =

6  / 24    =    1 / 4    =   25%

We break \(27^x\) into \((3^3)^x\)  , and  \(9^4\) into \((3^2)^4\) . Now, we have \(3^2*3^8=3^{3x}\) . Since the bases are equal, we have \(3x=2+8, 3x=10, x=\boxed{\frac{10}{3}}\)

1)The volume of a sphere is numerically equal to half its surface area. What is the radius of the sphere?

2)A plane cuts through a sphere with diameter 20 cm, but the closest it gets to the center is 3 cm. What is the area of the intersection of the sphere and the plane in sq cm?

asdfdasfasdf

1)

Volume =4/3 x Pi x r^3

Surface Area =4 x Pi x r^2

1/2[4 x Pi x r^2] =4/3 x Pi x r^3, solve for r

Solve for r:

2 π r^2 = (4 π r^3)/3

Divide both sides by π:

2 r^2 = (4 r^3)/3

Subtract (4 r^3)/3 from both sides:

2 r^2 - (4 r^3)/3 = 0

Factor r^2 and constant terms from the left hand side:

-2/3 r^2 (2 r - 3) = 0

Multiply both sides by -3/2:

r^2 (2 r - 3) = 0

Split into two equations:

r^2 = 0 or 2 r - 3 = 0

Take the square root of both sides:

r = 0 or 2 r - 3 = 0

Add 3 to both sides:

r = 0 or 2 r = 3

Divide both sides by 2:

r = 0-Discard                 or               r = 3/2

Solve for G:

Multiples of 3: 100/3= 33 1/3 - 33

Multiples of 4- 25, some are multiples of 3, so 25-8=17, so 33+17=50

Wait, so the total is 60, right? Okay \( x^2-2x-x+2x+8-x+x+10=60\)

Take its reciprocal:

=1 / 225

x= .0044444......

10x = .04444....

10x - x =  .04444.....   - .004444.... 

      9x = .04

       x = .04/9

          = 4/900    or 1/225 simplified. 

Hi Linx7    

\(\frac{3x-2}{4}-\frac{2x-5}{3}=\frac{1+x}{6}\\ \text{lowest common denominator is 12}\\ 12\times \left(\frac{3x-2}{4}-\frac{2x-5}{3}\right)=12 \times \left(\frac{1+x}{6}\right)\\ 3(3x-2)-4(2x-5)=2(1+x)\\ 9x-6-8x+20=2+2x\\ x+14=2x+2\\ 12=x\\ x=12\)

What about n=34?

(n*3=102, 2m=48*n+102=16*3*34+102=

16*102+102=102*17

HCF of n3 and 2m is 102

2sqrt2 is an irrational number and cannot be written exactly in digits.

Your notation is not correct.

What is the little 5 for?

\(x(x-2)+x+2x+8+10=60\\ x^2-2x+3x+18=60\\ x^2+x-42=0\\ (x-6)(x+7)=0\\ \text{x has to be positive so it is 6}\)

Number that are british is x+2x+8 = 3x+8 = 18+8 = 26

\(P(british)=\dfrac{26}{60}=\dfrac{13}{30}\)

EDIT:    Sorry this is not  right

I overlooked the fact that the painting must be from the 20th century (T) to start with.

There are 30 in from the 20th century

6 of those are british

P( that it is british given that it IS from the 20th century) = 6/30 = 0.2

(sqrt(-2(h+x))-sqrt(-2x))/h

\(\frac{{\sqrt{-2(h+x)}-\sqrt{-2x}} }{h}    \\ \frac{{\sqrt{-1}\sqrt{2(h+x)}-\sqrt{-1}\sqrt{2x}} }{h}    \\ \frac{\sqrt{-1}\left[\sqrt{2(h+x)}-\sqrt{2x}\right]}{h}    \\ \frac{\left[\sqrt{2h+2x}-\sqrt{2x}\;\right]\;i}{h}    \\ or\\ \frac{\left[\sqrt{h+x}-\sqrt{x}\;\right]\sqrt2\;i}{h}    \\ \)

the final answer really depends on what you are doing it for.

Let m and n be positive integers such that m = 24n + 51. What is the largest possible value of the greatest common divisor of 2m and 3n?

m=3(8n+17)

Well

if n is odd the HCF is 3

if n is even the HCF is 6

Charlie the Chimp is taking a multiple choice test.

There are four questions on the test and there are three answer choices for each question.

Only one answer choice is correct for each question.

Since Charlie is a chimp, he will be guessing on each question. As a common fraction, what is the probability that he guesses exactly one right answer?

4 independent trial. Probability of a correct answer each time is 1/3 so

P(exactly one correct)

\(= \binom{4}{1} * (\frac{1}{3})^1 * (\frac{2}{3})^3\\ =4 * \frac{1}{3}*\frac{8}{27}\\ =\frac{32}{81}\\ \)

2 sqrt 2   IS a number...... = 2.828427125

Can you clarify your question???

Not sure about this.....

x = lower tix

591 - x = upper tix

x(9.5) + (591 - x)(8.0) = 5281.5      solve for x = 369 

so  

Lower tix = 369 

   Upper tix = 591-369= 222 tix

1-100

100/3 = 33

100-33 =67

100/4 = 25     BUT ...some of those are multiples of 3     12 24 36 48 60 72 84 96 and have already been counted as multiples of 3    

                         so instead of 25   it is    25 - 8 = 17

33+17 = 50

100-50 = 50 numbers left   

This is true no matter how many roots there are. 

Oh wait, er I think I was overthinking a bit too much. Basically I think for the four questions, there are four cases. 

To get the first question correct, and the rest of the questions need to be wrong, so the probability would be 1/3*2/3*2/3*2/3 or 8/81. I think I would multiply this by 4 because the question you get correct can be any question. So the answer is 32/81.....yay?

The probability of drawing two red candies without replacement is 13/35 ,  and the probability of drawing one red candy is 2/5 .

What is the probability of drawing a second red candy, given that the first candy is red?

\(\frac{2}{5 }x=\frac{13}{35}\\ x=\frac{13}{35}\div \frac{2}{5 }\\ x=\frac{13}{35}\times \frac{5}{2 }\\ x=\frac{13}{14} \)