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For b., if a were 13 then there is one arrangement where all of them are 13, if a were 12 then there are 4 arragements... I wonder if there is a pattern.

I did get 715 for the first part, and I did not have trouble with that. For part b, I just don't know whether to approach this problem with casework or some sort of counting technique like stars and bars. One of the hints for part b was to approach the problem by designating variables.

Yeah I just saw that...

Yes, it is 13 C 4.... 0 and 12 are included...

I checked the answer to that question and the answerer did not consider 0 as a valid digit, therefore all of his/her answers were based off of 12 instead of 13.

There would be 13 integers to choose from right?

So it would be 13 C 4= 715

The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is
the length in meters of a diagonal of the rectangle?

Let's say that the length is x and the width is y, then the perimeter is 2x + 2y, and making that into an equation we get...

2x + 2y = 56

Simplifiying we get...

x + y = 28

The ratio of the length and width is 4 to 3, so we can make the equation...

4x = 3y

3(x + y) = 28

3x + 3y = 84

Plugging in 3y as 4x we get 

3x + 4x = 84

7x = 84

x = 12

Now we plug x = 12, into the original equation:

x + y = 28

x = 12

y = 28 - 12 = 16

The question asks us to find the diagonal, and that is going to be using the Pythagorean theorem. 

In this case, the hypotenuse is the diagonal of the triangle if we split the rectangle along the diagonal. 

a^2 + b^2 = c^2

12^2 + 16^2 = 400

c^2 = 400

c = 20

Since we know that the hypotenuse (c) is diagonal, and that is what the question is asking for it is our answer.

Answer: 20 meters

(a): This seems complicated, yet I think 13 C 4 = 715 ways if a is the least, b is the second to least, etc.

Yes, you're right...

lol I was the guest who posted above you :D

Perimeter  = 2 ( W +L)

56  =  2 (W + L)

And 

L / W  = 4/3   

L = (4/3)W

So we have  that

56  = 2  ( W + (4/3)W )     divide both sides  by 2

28  =  W ( 1 + 4/3)

28  = W( 7/3)     multiply both sides by  3/7

28 ( 3/7)  = W

(28/7) * 3  =  W

4 * 3  = W

12  = W

And  L = (4/3)(12)  = (12/3) * 4 = 4 * 4  =16

And  the  length  of  the diagonal  =√ [ W^2 + L^2 ] =  √[ 12^2 + 16^2 ] = √[144 + 256]  = √400 = 20 m

The keywords noted here are different=distingusiable balls and boxes(you can tell the balls apart)...

Let's say the for balls have color white, yellow, red, and blue... and the boxes are labeled from box 1, box 2, box 3, and box 4.

Pick up the first white ball, I can place it in any of the four boxes, for 4 ways.

Next, pick up the yellow ball, I can place it in any of the four boxes again(since there is no restriction one ball per box)..

Similarly, the same flow for the red and blue balls..

Thus, the answer is 4*4*4*4=4^4=256 ways.

The choices are independent of each other...

                       Length is   4/7     width is     3 / 7

                                        32                        24   

EACH side is                   16                        12

Diagonal (via Pythagorean theorem)     sqrt (16^2  + 12^2)   

If the perimeter of the rectangle is 56, that means that the length + the width equals 28. Since the ratio of the sides is 4:3, we can find that the length is 16 and the width is 12. From there we use Pythagorean thereom to find the length of the diagonal. 12^2+16^2= 400= 20^2. So the length of tge diagonal of the rectangle is 20 meters.

Hope it helps!

The path  length  is just  the circumference  of a circle  with a radius of 7/2  =  3.5ft

So.....the path  lebgth =

2 pi * radius  =

2 * 3.14 * 3.5  =   21.98 ft

The neat thing about this process is that you will always get -30 as your answer.

a. If you have a number x, then you do all the steps, you'll get:

x

x-10

6(x-10)

(6(x-10)/2

(6(x-10)/2-3x...this would be an unsimplified rule.

b. To prove that all numbers make the final result -30, we simplify the rule...

x

x-10

6(x-10)= 6x-60

(6(x-10)/2= 3x-30

(6(x-10)/2-3x= 3x-30-3x= -30

This proves that all inputs would create the same -30 output.

*I'm not sure if this fits your problems induction and deductive reasoning requirments, but I hope it gives you an idea on how to do so.

Hope it helps!

Thanks!

@HELPMEEEEEEEEEEEEE here you go :)

Diagram:                                                                                                                                                                                    

Note that \(\overline{AP}\) is a chord of both circles. Since both circles have the same radius, chord \(\overline{AP}\) must subtend the same angle, i.e. \(\angle ABP = \angle ACP\). Thus, triangle \(ABC\) is isosceles with \(AB=AC\).

Let \(M\) be the midpoint of \(\overline{BC}\). We see that\(BC = BP + PC = 14 + 4 = 18\) , so \(BM = CM = BC/2 = 9\). Then \(PM = CM - CP = 9 - 4 = 5\). Since \(\overline{AM}\) is also an altitude, we can apply Pythagoras to right triangle \(AMP\) to get \(AM=12\).

Hence, the area of triangle \(ABC\) is \(1/2 \cdot AM \cdot BC = 1/2 \cdot 12 \cdot 18 = \boxed{108}\).

Will do!

I understand that you can do that but if you want an EXACT answer, then pi/4 would be the answer required.

I've already explained how to do this question when you asked it in exactly the same way earlier. If you still need help, you might want to see what's missing from the question.

Thanks!

or you can do the radius*pi

80% of 35 is 28 and 25% of 28 is 7. That means 35 people have been memvers or more then 2 years. 35/63=5/9. The answer is 55 5/9%