MATH SOLVE

3 months ago

Q:
# In the xy-plane, C and D are circles centered at the origin with radii p 17 and p 5, respectively. Quantity A: The number of points (a; b) on circle C where both a and b are integers Quantity B: The number of points (a; b) on circle D where both a and b are integers A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Accepted Solution

A:

Answer:Quantity A is greaterStep-by-step explanation:Well, to solve this we'll do it by parts.Quantity A) 809 coordinate pointsSince it's centered at the origin we can write the circle equation as[tex]x^{2} +y^{2} =17^{2}We can rewrite it as:[tex]x^{2} +y^{2}=289[/tex]For now, we set apart the following points (0,17) (0,-17), (17,0) and (-17,0) because those points already belong to the circumference and circle.Now we're going to enlist the cartesian points. The parameter is the Radius. No point can be outside the circle, i.e. with a distance longer than 289 units. Doing it with the Quadrant I. We'll enlist them. And test by squaring(x,y) x²+y²≤289(0,0) Origin(1,1); ...(1,16) (1,16)∈ for 1²+16² ≤289(1,1); ...(1,16) = 16 points(2,1)....(2,16) =16 points (7,1)...(7,15) =15 points (12,1)...(12,12)=12 points(3,1)....(3,16) =16 points (8,1)....(8,15) =15 points (13,1)...(13,10)= 10 points(4,1)....(4,16) =16 points (9,1)...(9,14) =14 points (14,1)...(14,9)= 9 points(5,1)....(5,15) =15 points (10,1)...(10,13) =13 points (15,1)...(15,8) =8 points(6,1)...(6,15) =15 points (11,1)...(11,12)= 12 points (16,1)...(16,5) =5 pointsSo 16+16+16+15+15+15+15+14+13+12+12+10+9+8+5=48+60+27+24+27= 186 coordinate points for the Quadrant I186*4=744 coordinate points for all four Quadrantsy axis points: (0,1)..(0,16) =16 points (0,-1)...(0,-16)=16 points x-axis points (1,0)...(16,0)=16 points (-1,0)...(-16,0)=16 pointsOrigin (0,0)= 1 coordinate pointNotice that I'm only counting the points on the circle not on the circumference, as the question asks.Adding it all up:744 +16+16+16+16+1=809 coordinate points2) Quantity B: 77 coordinate pointsSimilarly:x²+y²=5²x²+y²=25Points on the y-axis: 8 coordinate points. Points on x-axis: 8 coordinate points(1,1)...(1,4) =4 points (2,1)...(2,4) =4 points (3,1)..(3,4)=4 points (4,1)...(4,3) =3 pointsIn Quadrant I = 15In 4 quadrants = 60 coordinates points60 coordinates points +4+4+4+4+1=77 coordinate pointsFinal comments:We could solve it this only estimating. Just based on the radii of those circles. Since the question does not ask any quantity, but just the relationship. This could be a time saver way.