Let G be the graph of the parametric equations
x = cos(4t),
y = sin(6t).
What is the length of the smallest interval P such that the graph of these equations for all t E P produces the entire graph G?
if 4 t = 2 pi
t = pi/2
if 6 t = 2 pi
t = pi/3
so we need at least t = 0 to t = pi/2 or any other interval of length pi/2
In the interval -pi/4 <= t <= pi/4, the graph is traced once.
But on 0 <= t <= pi/2, only half the curve is completed.
Try changing the domain on this plot at
http://www.wolframalpha.com/input/?i=plot+x%3Dcos(4t),y%3Dsin(6t),+-pi%2F4+%3C%3D+t+%3C%3D+pi%2F4
To find the length of the smallest interval P such that the graph of the parametric equations produces the entire graph G, we need to determine the range of t-values for which the x and y components of the equations cover the entire graph.
Let's start by examining the x-component of the parametric equations, x = cos(4t). The cosine function has a period of 2π, which means that it repeats every 2π units. Therefore, the x-component will complete one full cycle (covering the entire range of values between -1 and 1) when 4t completes one full cycle. This occurs when 4t covers an interval of 2π. Thus, the period of the x-component is given by:
Period of x = 2π / 4 = π/2
Similarly, we can analyze the y-component of the parametric equations, y = sin(6t). The sine function, like the cosine function, has a period of 2π. However, in this case, the y-component will complete one full cycle when 6t completes one full cycle. Thus, the period of the y-component is given by:
Period of y = 2π / 6 = π/3
Now, to find the smallest interval P for which the graph of the parametric equations produces the entire graph G, we need to find the least common multiple (LCM) of the periods of the x and y components. The LCM of π/2 and π/3 can be calculated as:
LCM(π/2, π/3) = π * 2 * 3 / gcd(2, 3) = 6π / 1 = 6π
Therefore, the length of the smallest interval P is 6π.
To summarize:
The length of the smallest interval P such that the graph of the parametric equations for all t E P produces the entire graph G is 6π.
To find the length of the smallest interval P that produces the entire graph G, we need to determine the period of both x = cos(4t) and y = sin(6t).
The period of x = cos(4t) is given by T1 = 2π/4 = π/2. This means that the graph of x = cos(4t) repeats every π/2 units along the t-axis.
Similarly, the period of y = sin(6t) is given by T2 = 2π/6 = π/3. This means that the graph of y = sin(6t) repeats every π/3 units along the t-axis.
To find the smallest interval P, we need to find the least common multiple (LCM) of T1 and T2.
LCM(π/2, π/3) = π * (2/2)(3/3) = π.
Therefore, the length of the smallest interval P is π.