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 ?
To find the number of lattice points on the graph of the given parametric equations, we can substitute the equations into the range given for x and y and solve for t.
Given:
x = 30 - 40cos(t)
y = -50 + 30cos(t)
-100 ≤ x ≤ 100
-100 ≤ y ≤ 100
Substituting x and y equations into the given range:
-100 ≤ 30 - 40cos(t) ≤ 100
-100 ≤ -50 + 30cos(t) ≤ 100
Now, we can solve these inequalities separately:
For x equation:
-100 ≤ 30 - 40cos(t) ≤ 100
We can rearrange this inequality to isolate cos(t):
-130 ≤ -40cos(t) ≤ 70
Divide all the terms by -40 (since -40 is negative, we must flip the inequality signs):
3.25 ≥ cos(t) ≥ -1.75
Cosine(t) can take values between -1 and 1, so the range of cos(t) is [-1, 1]. Therefore, we can limit the range to the given interval:
-1 ≤ cos(t) ≤ -0.54
Now, let's solve the second inequality for y equation:
-100 ≤ -50 + 30cos(t) ≤ 100
Add 50 to all the terms to isolate cos(t):
-50 ≤ 30cos(t) ≤ 150
Divide all the terms by 30:
-1.66 ≤ cos(t) ≤ 5
Again, since cosine(t) can take values between -1 and 1, we limit it to the given interval:
-1 ≤ cos(t) ≤ 1
Now, we need to find the intersection of both ranges for cos(t):
The first inequality for cosine(t) is -1 ≤ cos(t) ≤ -0.54. The only values in this range are between -1 and -0.54, which means they do not intersect with the second range -1 ≤ cos(t) ≤ 1.
Therefore, there are no points (x, y) with integer coordinates on the graph of the parametric equations within the given ranges.