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Math Help - find number of numbers satisfying the following inequations

  1. #1
    Senior Member nikhil's Avatar
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    Exclamation find 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
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: 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.
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  3. #3
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    Re: find number of numbers satisfying the following inequations

    Hello, nikhil!

    How many natural numbers satisfy these inequations?

    . . x+y \:\ge\:7
    . . x^2+ y^2\:\le\:30

    We are seeking points on or above the line x + y \:=\:7
    . . and points on or inside the circle x^2+y^2\:=\:30.

    Since x and y are natural numbers,
    . . we are seeking "lattice points" only.

    I found four points: . (2,5),\,(5,2),\,(3,4),\,(4,3).
    Thanks from phys251
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  4. #4
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    Re: 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.
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