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wait how can t be so big. My answer choices are 40,55,60,68,85. I think it’s 68

Yeah, I think the Latex is not showing up.

No problem!

Oh, sorry... 98.......198=49+2=51 multiples of two and 99.....198....33...66, 34 multiples of three.  We have to subtract the multiples of 6, 102....198, 17..33, 33-16=17. 51+34-17=68.

No numbers appear....!!!

sin x =  sqrt (2)  - 1

sin^2 x = [ sqrt (2) - 1 ] ^2  = [ sqrt (2) - 1 ] * [ sqrt (2) - 1 ] =  

sqrt (2) *sqrt (2) - 1*sqrt(2) - 1*sqrt(2) + (-1)(-1)  =

2 - 2sqrt (2) + 1  =

[ 3 - 2sqrt (2) ]

So.....using an identity.....

cos^2x = 1 - sin^2 x

And we have that

1 - [ 3 - 2sqrt(2) ] =

-2 + 2sqrt (2)   .....so   a = -2   and b = 2 

The third important property of triangles is the triangle inequality rule, which states: the length of a side of a triangle is less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.

so let 10 = a  

10 < b+c         well.... a good integer value for b+c   would be 11

10 >  b-c           If we let b be the bigger of the two    10 and 1     would work for b and c respectively  ....or    7 and 4   or 6 and 5    or 6.5  4.5 etc

Anyway   10 + 11 = 21

Perhaps you're looking for something like 5(2a-5+b)?

Guest, I am not sure why the answer key would exclude the other valid solutions. 

Let James money = x

Let his sister's money  x + 27

Let his brother's money = 2x

After the payoff....

James brother has 2x - 24

And James sister has   x + 27 + 12  =  x + 39

So

x + 39 =  2 [ 2x - 24 ]

x + 39 = 4x - 48

87 = 3x

x = 29

So

James has  29 + 12   = $41

We can simplify this as

x / [ x - 10 ]  > [ y - 5] / y  

Since  0 < x, y < 10.....the left side will always be negative

And the right side will be negative when y < 5

So  when x = 1

The possible integer values for y are 1, 2, 3 and 4

When x = 2

The possible integer values are 1, 2, 3

When x = 3

The possible integer values of y are  1,2,3

When x = 4

The possible integer values for y are  1, 2

When x = 5

The possible integer values for y are 1,2

When x = 6

The possible values for y are 1

When x = 7

The possible values for y are 1

When x = 8...there is no value of y that makes the inequality true

So....there are 16 pairs of (x, y)  that make the inequality true

Again....if we cannot access your question....we cannot solve it.....this site requires a login which makes it inaccessible

Let's use    "x" for "a"   and "y" for b

We can solve this with a graph

The region created by the inequalities will be a 4 x 6  rectangle = 24 units^2

But the only values of (x, y) that produces a positive product for xy    will be  where x and y have the same signs

These areas will consist of a 3 x 2  rectangle and a 4 x 1 rectangle

So...the total of these areas is 6 + 4 = 10

So....the probability that  xy =  ab is positive is   10 /24   =  5 /12

Here's a graph to give you a feel for this :

https://www.desmos.com/calculator/834uyfbxoo

got it! thanks so much

please? i swear ill make an account for the next one

This site wants a login.....

The easiest way to see this is to imagine that we can "unfold" the spiral to create a rectangle

The bottom side of the rectangle is just the arc length of a circle of a radius of 3 ft  turned through 270°= (3/4) of a whole revolution....in other words.....(3/4) (2pi) (3)  =  (9/2)pi ft =   4.5 pi ft

The side of the rectangle will be rise of the staircase = 10ft

And the diagonal of the rectangle will be the length of the handrail

We can use the Pythagorean Theorem to solve this

handrail length = √ [ (4.5 pi)^2 + 10^2 ]  ≈  17.3 ft

1/15   chance of eclair    

1/15  =   3 / (3 +  2x+6  + x)

1/15 = 3/(3x+9)     cross multiply

3x+9 = 45

x = 12

then the number of humbugs = 2x+6 = 2(12) + 6 = 30 humbugs

Prob of humbug   =   30 humbugs /(total sweets)

30 / ( 3+ 30 +12) = 30/45 = 2/3 prob of humbug

Thank you!

1 + sin xcos^2x = sinx

We need to be careful about dividing through by a trig function, YEEEEEET...we might be throwing away a possible solution!!!

We can wrtite this as

1 + sin x ( 1 - sin ^2 x)  = sin x

1 +sin x  - sin ^3 x = sin x

1 - sin^3 x =  0

sin^3x  - 1 = 0    factor as a difference of cubes

(sin x - 1) (sin^2x + sin x + 1) = 0      

The second factor produces no real solutions for   x

For the first factor

sin x - 1 = 0

sin x = 1

And this happens when x = 90°

The graph here confirms this

https://www.desmos.com/calculator/lljppoztzo

Surface area of a sphere =4*Pi*R^2
Surface area of a cube =S^2 x 6
2^2 x 6 =24 - SA of the cube.
24 = 4*Pi*R^2, solve for R
R = Sqrt(6/pi)
Volume of sphere =4/3 x Pi x [sqrt(6/pi)]^3
4/3 x Pi x [sqrt(6/pi)]^3 =K*sqrt(6) /sqrt(pi), solve for K
K = 8

Area of the cube =   2 x 2 *6 = 24 sq units  

VOLUME of a sphere = 4/3  pi r^3     find r from the volume equation given

4/3 pi r^3 = k sqrt 6 / sqrt pi

r^3 = 3/4  * k sqrt 6 /  pi sqrt pi

r = cubrt ( 3/4 *k sqrt 6/ (pi sqrt pi )

Surface of a sphere = 4 pi r^2     sub in the r value and equate to 24 sq units

4 pi (cubrt 3/4 *k sqrt 6/ (pi sqrt pi))^2 = 24

Solve fo k = 8