A lattice point is a point with integer coordinates. How many lattice points $(x,y)$ with $-100\le x\le 100$ and $-100\le y\le 100$ are on the graph of the parametric equations
\begin{align*}
x&=30-40\cos t,\\
y&=-50 + 30\cos t?
\end{align*}
21
Well, let's just say that graph is a bit sinusoidal... or sinusoy-lattice-al, if you prefer.
To figure out how many lattice points there are, we need to figure out how many values of $t$ give us integer values of $x$ and $y$.
Let's tackle the $x$ equation first. We need $x$ to be an integer, so we need $30 - 40\cos t$ to be an integer. Now, $\cos t$ cycles between $-1$ and $1$, so in order for $30 - 40\cos t$ to be an integer, $40\cos t$ needs to be an integer.
That means $\cos t$ needs to be a rational number, which happens for certain values of $t$.
For $y$ to be an integer, we need $-50 + 30\cos t$ to be an integer. Similarly, to make things simpler, we need $30\cos t$ to be an integer.
Combining these two conditions, we see that $40\cos t$ and $30\cos t$ both need to be integers. Well, the only numbers that are both multiples of $40$ and $30$ are... multiples of $120$, of course!
So for every multiple of $120$ degrees (or $\frac{2\pi}{3}$ radians), both $x$ and $y$ will be integers.
Now, the $x$-coordinates span from $-100$ to $100$, so we have a total of $201$ possible $x$-values. For each multiple of $120$ degrees, the corresponding $x$-coordinate is an integer.
Since $\frac{360\text{ degrees}}{120\text{ degrees}} = 3$, we have a total of $3$ possible $x$-values for each multiple of $120$ degrees.
Therefore, the total number of lattice points on the graph is $201 \times 3 = 603$.
And there you have it! The lattice points are having a trig-tastic party on that graph.
To find the lattice points on the given graph, we need to find values of t that will yield integer values for x and y.
First, notice that the range of $x$ is $-100\le x\le 100$, so we can set up the inequality $-100 \le 30 - 40\cos t \le 100$ and solve for $t$.
\begin{align*}
-100 &\le 30 - 40\cos t \le 100 \\
-130 &\le -40\cos t \le 70 \\
\frac{13}{4} &\le \cos t \le -\frac{7}{4}
\end{align*}
Since $\cos t$ is bounded between -1 and 1, the inequality is only satisfied when $\cos t$ is equal to 1 or -1.
When $\cos t = 1$, we have $x = 30 - 40 = -10$.
When $\cos t = -1$, we have $x = 30 - (-40) = 70$.
So we have two possible values for x, -10 and 70.
Now, let's repeat the process for y. The range of $y$ is also $-100\le y\le 100$, so we can set up the inequality $-100 \le -50 + 30\cos t \le 100$ and solve for $t$.
\begin{align*}
-100 &\le -50 + 30\cos t \le 100 \\
-50 &\le 30\cos t \le 150 \\
-\frac{5}{3} &\le \cos t \le 5
\end{align*}
Again, since $\cos t$ is bounded between -1 and 1, the inequality is only satisfied when $\cos t$ is equal to 1 or -1.
When $\cos t = 1$, we have $y = -50 + 30 = -20$.
When $\cos t = -1$, we have $y = -50 + (-30) = -80$.
So we have two possible values for y, -20 and -80.
Therefore, there are four lattice points on the graph of the parametric equations: (-10, -20), (-10, -80), (70, -20), and (70, -80).
To determine the number of lattice points on the graph of the given parametric equations, we need to find all possible values of $t$ for which the coordinates $(x,y)$ are integers within the given range.
We can start by rewriting the equations in terms of $t$:
\begin{align*}
x&=30-40\cos t \implies \cos t = \frac{30-x}{40},\\
y&=-50 + 30\cos t \implies \cos t = \frac{y+50}{30}.
\end{align*}
Since the cosine function has a range of $[-1,1]$, we know that both $\frac{30-x}{40}$ and $\frac{y+50}{30}$ must be within this range. This gives us two inequalities:
\begin{align*}
-1\le \frac{30-x}{40} \le 1 \implies -40\le 30-x \le 40 \implies -70 \le x \le 90,\\
-1\le \frac{y+50}{30} \le 1 \implies -30\le y+50 \le 30 \implies -80 \le y \le -20.
\end{align*}
Since $-100\le x\le 100$ and $-100\le y\le 100$, we can restrict our attention to the integer values of $x$ and $y$ that fall within the above ranges. So, we have:
\[x\in\{-70,-69,\ldots,90\} \quad \text{and} \quad y\in\{-80,-79,\ldots,-20\}.\]
Now, for each value of $x$, we need to determine the corresponding values of $t$ that satisfy the equation $\cos t = \frac{30-x}{40}$. To do this, we apply the inverse cosine function on both sides:
\[t = \arccos\left(\frac{30-x}{40}\right).\]
Using this expression, we can find the values of $t$ for which the coordinates $(x,y)$ are integers. We do the same for $y$ using the equation $\cos t = \frac{y+50}{30}$:
\[t = \arccos\left(\frac{y+50}{30}\right).\]
By finding all the possible values of $t$ that we obtain from these equations, the number of lattice points on the graph of the parametric equations can be determined.