Let G be the graph of the parametric equations x=cos(4t)and y=sin(6t). What is the length of the smallest interval I such that the graph of these equations for all t in I produces the entire graph G?

the period for x is pi/2

the period for y is 2pi/3

So, I'd say I=[-pi/3,pi/3]

That's wrong :(

It's not 2pi/3 or pi

To find the length of the smallest interval I that produces the entire graph G, we need to determine the period of the functions x = cos(4t) and y = sin(6t), and then find the least common multiple (LCM) of these periods.

The period of a trigonometric function is the smallest positive value of T for which the function repeats itself. For the function x = cos(4t), the period is given by Tx = 2π/4 = π/2. Similarly, for the function y = sin(6t), the period is Ty = 2π/6 = π/3.

Now, we find the LCM of the two periods, Tx and Ty. The LCM is the smallest positive value that is evenly divisible by both periods. To find the LCM, we can factorize the periods into their prime factors:

Tx = π/2 = (2 * π * 1) / (2 * 1) = π * 1
Ty = π/3 = (2 * π * 1) / (3 * 1) = (2/3) * π * 1

Now, we identify the common prime factors and include the highest power of each factor:

LCM = (2/3) * π * 1 = (2/3)π

Therefore, the smallest interval I that produces the entire graph G is given by I = (0, (2/3)π). The length of this interval is |I| = (2/3)π - 0 = (2/3)π.

So, the length of the smallest interval I is (2/3)π.