A lattice point is a point with integer coordinates. How many lattice points (x,y) with -100 less than or equal to x less than or equal to 100 and -100 less than or equal to y less than or equal to 100 are on the graph of the parametric equations x=30-40 cos t

Question

A lattice point is a point with integer coordinates. How many lattice points (x,y) with -100 less than or equal to x less than or equal to 100 and -100 less than or equal to y less than or equal to 100 are on the graph of the parametric equations x=30-40 cos t
y=-50 + 30 cos t ?

Erm....looks really complicated, how do I approach this? Do I solve for t in terms of x and plug into the other one and try to do some inequality work?
Thanks

Reiny · Answer

x = 30 - 40cost ---> cost = (30 - x)/40
y = -50 + 30cost --- cost = (y+50)/30

(y+50)/30 = (30-x)/40
40y + 200 = 900 - 30x
30x + 40y = 700

3x + 4y = 70

So your graph is simply a straight line with
x -intercept of 70/4 or 17.5 and
y-inercept of 70/3 or 23 1/3
draw the line for your given constraints.

Now can you figure out a way to find how many lattice points , or points with integer numbers, are in your triangle ?

Sam · Answer

Thanks, but why is it a triangle and not a line I graphed it and it is a line do I have to connect the ends and stuff?

Leyla · Answer

It is the line, but with a negative slope and a different intercepts, equation is 3x + 4y = -110 (50 multiplied by 40 is 2,000, not 200 :-) )