hi,

got stuck on a rather easy looking question. How many natural numbers satisfy the inequations

1) x+y >= 7

2) (x^2)+(y^2) <= 30

thanks

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- Jan 21st 2014, 10:54 AMnikhilfind number of numbers satisfying the following inequations
hi,

got stuck on a rather easy looking question. How many natural numbers satisfy the inequations

1) x+y >= 7

2) (x^2)+(y^2) <= 30

thanks - Jan 21st 2014, 11:03 AMebainesRe: find number of numbers satisfying the following inequations
I suggest trying combinations of numbers and seeing what works. "Natural numbers" are positive integers, although some people define them to include the number 0 as well. So let's try:

Case 1: Let x+y = 7. Possible combinations for x & y are: 0 & 7, 1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1, 7 & 0. Of these, which ones (if any) have the property that x^2 + y^2 <=30?

Case 2: now try x + y = 8. Posible combinations for x & y are: 0 & 8, 1 & 7, 2 & 6, 3 & 5, 4 & 4, etc. Again, which if any of these have the property that x^2 + y^2 <=30?

You should be able to see that the sum of the squares is smallest if x=y, which means for x+y = N that x and y both equal N/2. So once N gets large enough that (N/2)^2 > 30 there's no point in checking larger numbers. - Jan 22nd 2014, 04:00 PMSorobanRe: find number of numbers satisfying the following inequations
Hello, nikhil!

Quote:

How many natural numbers satisfy these inequations?

. . $\displaystyle x+y \:\ge\:7$

. . $\displaystyle x^2+ y^2\:\le\:30$

We are seeking points on or above the line $\displaystyle x + y \:=\:7$

. . and points on or inside the circle $\displaystyle x^2+y^2\:=\:30.$

Since $\displaystyle x$ and $\displaystyle y$ are natural numbers,

. . we are seeking "lattice points" only.

I found four points: .$\displaystyle (2,5),\,(5,2),\,(3,4),\,(4,3).$ - Mar 5th 2014, 03:47 PMmathguy25Re: find number of numbers satisfying the following inequations
x + y => 7

x^2 + y^2 <= 30

If x or y is > sqrt(30) > sqrt(25) = 5, then y^2 will exceed 30. Thus, consider points (x, y) where 1 <= x, y <= 5

(1, 1) (2, 1) (3, 1) (4, 1) (5, 1)

(1, 2) (2, 2) (3, 2) (4, 2) (5, 2)

(1, 3) (2, 3) (3, 3) (4, 3) (5, 3)

(1, 4) (2, 4) (3, 4) (4, 4) (5, 4)

(1, 5) (2, 5) (3, 5) (4, 5) (5, 5)

Eliminate the ones that don't satisfy your equation.