Thanks, Melody......time constraints prevent me from explaining this in more detail......

(x-2)(x²-1)>0

Melody, I think they want the sum  of  [5N + 3]  for  N =  1 to 70...so we have.....

5[ (70)(71) / 2] + 70 (3)  = 12635

He averages  one mile evey 17 minutes.....thus....it would take him 34 minutes to walk two miles

Remember, Dragonlance, x = 3 isn't a  part of the solution.......I'm just doing this to find the interval points.......the solution is x > 3 as sabi92 found ....  or....as I gave ..... (3, ∞ )......both of these are the same solutions

Here's the way I like to approach this kind of problem....

Solve for the top and bottom separately....

Set the top  = 0   and factor....we have....   x^2  - 4x + 4  = 0   → (x - 2)^2 =  0    so   x =2

Do the same for the bottom

x - 3 = 0      so x = 3

Now.....plot these two values on a number line......neither will be part of the answer (because of the > sign), but the answer(s)  will come from one or more of these intervals.....

(-∞, 2), (2, 3)  or (3, ∞)

Pick a number in the first interval....0 seems nice ....put it nto the original problem.....does it make it true??...Nope

Pick a number in the middle interval...say 2.5......you will find this doesn't work, either

Finally.....pick a number in the last interval, say 4.....you will find that this is the only interval that solves this problem

A BIG HAPPPY B'DAY to both jillybean and Rose 98......

Maybe we'll skip the Trojan horses!!!

Here's another way to prove this using the Binomial Theorem

Let (30)^99  = (31 - 1)^99

Let  (61)^100  = (62 - 1)^100

So we have

[(99C0)(31)^99  -  (99C1)(31)^98 + (99C2)(31)^97 + ....+ (99C98)(31) - 1 ]

+

[(100C0)(62)^100 - (100C1)(62)^99 + (100C2)(62)^98 + ..... +(100C98)(62)^2 - (100C99)(62) + 1 ]

And adding these, the last terms cancel, and every other term in both expressions is divisible by 31.....

9:12 is the same ratio as 3:4  but not the same as 9:16

I'm assuming that by "similar" you mean that the radiuses are in the same ratio, as well...3:4

So....the ratio of the volumes is just (3/4)^3 = [pi *3^3 *3] [pi * 4^2 *4] = 27/64

And the ratio of the surface areas = 2 pi [ (3)^2 + 3* 3] /2 pi [(4)^2 +4*4] =

[18 ] / [ 32] = 9/16