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thank you for the detailed explanation

ye i always struggle to get the slope for those kinds thank you!

deleted. Sorry, misread the "floor function"

I think you made a slight error, Melody

You computed (1/10) of the area of the circle...not (1/10) of the circumference

Ah...I see....Ant wanted the perimeter of the sector.....!!!

This should be

Sum of radii + arc we found

10 + 10 + 2pi = [20 + 2pi ] =  26.28 units

And you are correct, Ant....

We have

ac = 1/42  ⇒  c = 1 / (42a)    (1)

bd = 1/72 ⇒ b = 1/(72d)       (2)

a + b = 7/24       (3)

c + d = 16/63     (4)

42ac = 1

72bd = 1

24a + 24b  = 7

63c + 63d = 16     this system is a little sticky to solve, but note that

a = (1/6)   b = (1/8)   c = (1/7)  d = (1/9)    make the system true

So   ad + bc  =  (1/6)(1/9) + (1/8)(1/7) = (1/54) + (1/56) =

(54 + 56) / 3024

110/ 3024 =

55 / 1512

The numbers   a   and  b   must multiply to 15    and add to k

a   b        can be        added =k

1  15                             16

3   5                                8

-1  -15                           -16

-3   -5                              -8

all of the k's multiplied results in   16384

2

Cross multiply to get

x^2 +3x = -9x-36

x^2+12x+36=0

(x+6)(x+6) = 0    x = - 6

3

x^2 -12x +32 -12 = 0

    x^2-12x+20 = 0

    (x-10)(x-2) = 0     so x = 10 or 2

4

x^2-7x -98 =0    use quadratic formula to find x = 14  or -7

∑[(nC15 / nC17), n, 17, infinity]=272

Thanks CPhill, since you are not here, and ant101 has asked for clarifictaion, I will add my input.

Hi Ant101  

There are 360 degrees in a circle

so 36 degrees is one tenth of a circle and the radius of your circle is 10

so    the arc length is    \(\text{one tenth }*\pi*10^2 = 0.1*\pi*100=10\pi\)

NOW, your question is worded incorrectly and technically it does not make sense.

An arc is a part of a circle's circumference and it does not have a perimeter because it is not a closed curve.

  An arc has a length, which I have alredy found for you.

I think you are actually asking for the perimeter of the sector.

A sector is a part of a circle which includes 2 radii and the arc between them.  

   For instance, when you cut a cake you usually cut it into sectors.

so

The perimeter of the sector is       \(10\pi+10+10 = 20+10\pi\;\;units\)

-----------------------

My answer is very different from CPhill's.  This is because the question is worded incorrectly and he interpreted its meaning very differently from me.

The truest thing!

( I feel like this really applies to me!)

Thank you, CPhill! So, the perimeter is 10+10+6.28=26.28 units?

Oh, OK....the radius = 10

Arc length =  arc length of whole circle * (36/360) =

2pi (radius) * (36/360) =

2pi (10) * (1/10 ) =

2pi  units ≈  6.28 units

Kind of looks like this!

Thanks, CPhill! But, it was a regular arc in a circle.

Let x be the number

Subtracting 8 from this  =  x - 8

Multiplying this by 5 = 5(x - 8)

And this = 20

So....we have

5(x - 8)  = 20     divide both sides by 5

x - 8  =  4          add 8 to both sides

x = 12

OK....  Let 'x' = the number

He subtracts 8        x-8

Multiplies the result by 5       5(x-8)

And that = 20                         5(x-8) = 20    

Now solve for x

5(x-8) = 20

5x - 40 = 20

5x = 60

x = 12

F= mg    = 4(1000)(10) = 40000 kg-m/s^2 = 40000 N

W= F x d = 40,000(3) = 120,000 kg-m^2/s^2= 120,000 j

P= W/t = 120000/60 = 2000 j/s

Ant....I assume that we have a decagon [ regular polygon of 10 sides ]  inscribed in a circle??......if so....

The radius, r, of the circle can be found as

r / sin 72 = 10/ sin 36

r = 10sin72 / sin 36 =    5 (1 + √5 )

So....the perimeter of the arc   [ arc length of 1/10 of the circle ] is

2pi r / 10 = 

2pi [ 5 ( 1 + √5)]  / 10    =    pi *  (1 + √5) units  ≈ 10.17 units

Can we use that 36 degrees is 1/10 of 360?

OK.....hope that helped

Thank you, CPhill!

Thanks, tertre.....!!!!

Yeah.... I should have labeled the "65" as "Both"

The number of students who didn't take either = 

460 - (75 + 65 + 280 )  = 42

We aren't summing 75 + 30  [ remember that some of this "sum" is being "double-counted"]

We are just noting that     73/460 take only Science.....65/460 take Both and 280/460 take Math

So       [ 73 + 65 + 280 ] /  460 = 418/460   take only Math, only Science, or Both

And 42 /460 take neither

Hello! Here is a long way!

If you know the principle of Inclusion and Exclusion, this will lead to the size of their union, but in this case, we have to find the intersection! Therefore, we can find the size of union and subtract from the number of multiples of 4 and 6. There are \(\frac{170}{4}\approx42\) multiples of \(4\) and \(\frac{170}{6}\approx28\) multiples of \(6.\)The least common multiple of 4 and 6 is 12, so \(\frac{170}{12}\approx14\) . You could stop here, because this is your answer, but let's continue and show why this works. The union is just \(42+28-14=56\) and there are \(42+28\) total multiple of 4 and 6. Thus, the intersection is \(70-56=\boxed{14}\) . You could've stopped well before and found your answer, but here is just a long way.