Melody

\(a^2-1=(a+1)(a-1)\)

so the most likely possibilities are

\([x-(a+1)]^3\\ [x+(a+1)]^3\\ [x-(a-1)]^3\\ [x+(a+1)]^3\\ \)

I'd eliminate the last one becasue there won't be any minus signs.

I'd test the others.

I have the answer as I have drawn it to scale but I have not worked out a non-graphical solution.

It quite likely has something to do with similar figures but I really do not know

I let the central crossection of the sphere be  \(x^2+y^2=16^2\)

So the crossection of the top half is    \(y=\sqrt{256-x^2}\)

The table is at y=-16

So the cut is at y=8

Using Pythagoras you will find that the positive end of the cut is at  \(x=8\sqrt3\)

-------

The volume of the sphere section from x=0 to 8sqrt3  will be

\(V=\displaystyle \pi\int_0^{8\sqrt3}\;256-x^2\;dx\\ V=\pi \left[256x-\frac{x^3}{3}\right]_0^{8\sqrt3}\\ V=\pi \left[256*8\sqrt3 -\frac{8^3*3*\sqrt3}{3}\right]\\ V=\pi \left[4*8^3\sqrt3 - 8^3\sqrt3\right]\\ V=\pi \left[3*8^3\sqrt3 \right]\\ \)

The volume of the rectangular p[rism section from  x=0 to x=8sqrt3 will be

\(V=\pi \displaystyle \int_o^{8\sqrt3} 8^2dx\\ V=\pi \displaystyle \left[64x\right ]_o^{8\sqrt3} \\ V=64*8\sqrt3\;\pi\\ V=8^3\sqrt3\;\pi\)

Subtract

Difference in volume is   

 \(V=2*8^3\sqrt3\;\pi\\ V=1024\sqrt3\;\pi\)

This is top and bottom so you can divide by 2 but it is only half of the needed region so you have to multiply by 2

So that is the  volume of the top bit.

You should be able to do at least a bit more by yourself.     

You have not simplified your question, you have just written it properly.  As you should have done int the first place.

Latex coding is greast if you render it properly.  Otherwise it is just coding garbage.

I know that if x and the denominator are relatively prime then

x/9 will have  one number repeating

x/99 will have 2

x/999 will have 3 digits repeating

..

x/99999999  will have 8 digits repeating

Now

factor(99999999) = 3^2 *11 * 73 * 101 *137

So n is going to have to be a multiple of 1 or more of these factors

1/3 only has 1 repeating digit

1/9 only has 2 repeating digitsw

1/11 only has 2 repeating digits

1/33 only has 2 repeating digits

1/99 only has 2 repeating digits

1/73 = 0.01369863 with all 8 digits repeating.

So the smallest n is 73  

you are after the cost per shirt   cost / shirt      That will be rise/run which is the gradient.

So take the difference in the cost of the shirts  and divide it by the difference in the number of shirts.   

You can draw the graph, it might help you understand, buyt you do not have to in order to answer the question

Because I am a teacher.  I try not to be a cheat facilitator.

Snce you responded i will give more.

G(1)= -3

Look at the graph to find out why.

Next find G(-3)  

and so on. 

You are welcome. :)

Well,

what is G(1)  ?

Hi Proaop,

\(f(6)+6f(1-6)=1+36\\ f(6)+6f(-5)=37\qquad (1)\\~\\ f(-5)-5f(1--5)=1+25\\ f(-5)-5f(6)=26\\ -5f(6)+f(-5)=26\qquad (2a)\\ -30f(6)+6f(-5)=156\qquad (2b)\\~\\ now\;\;\;(1)-(2b)\\~\\ \text{I expect you can finish it.}\)