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
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 ?
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?
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 :-) )
To approach this problem, you can substitute the given parametric equations into the inequality conditions and solve for the parameter, t.
Let's start with the given parametric equations:
x = 30 - 40 cos t
y = -50 + 30 cos t
Now, let's substitute these equations into the inequality conditions:
-100 ≤ x ≤ 100
-100 ≤ y ≤ 100
Substituting x and y from the parametric equations, we get:
-100 ≤ 30 - 40 cos t ≤ 100
-100 ≤ -50 + 30 cos t ≤ 100
Now, let's solve these inequalities separately.
For the first inequality:
-100 ≤ 30 - 40 cos t ≤ 100
Subtracting 30 from all sides, we get:
-130 ≤ -40 cos t ≤ 70
Dividing all sides by -40 (and changing the inequality direction since we're dividing by a negative number), we get:
3.25 ≥ cos t ≥ -1.75
Next, we solve the second inequality:
-100 ≤ -50 + 30 cos t ≤ 100
Adding 50 to all sides, we get:
-50 ≤ 30 cos t ≤ 150
Dividing all sides by 30, we get:
-1.67 ≤ cos t ≤ 5
Now, we have the range of values for cos t. To find the number of lattice points, we need to determine how many integer values within these ranges meet the criteria.
The smallest value cos t can take is -1, and the largest value is 1. So, we can consider the range for cos t to be -1 ≤ cos t ≤ 1.
Within this range, there are only two integer values, -1 and 1.
To calculate how many lattice points (x, y) satisfy the given conditions, we can count each value of t that corresponds to these integer values of cos t. Since the parametric equations are periodic with a period of 2π, there are two values of t that satisfy the equation for each integer value of cos t.
Therefore, the total number of lattice points is 2 * 2 = 4.
So, there are four lattice points (x, y) with -100 ≤ x ≤ 100 and -100 ≤ y ≤ 100 on the graph of the given parametric equations.