A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the frequency of the system is 8/pi cycles per second, determine a trigonometric model that gives the position of the weight at time t seconds.

To find a trigonometric model that gives the position of the weight at time t seconds, we will use the equation of simple harmonic motion.

The equation for simple harmonic motion can be expressed as:

x(t) = A * cos(ωt + φ)

where x(t) is the position of the weight at time t seconds, A is the amplitude of the motion, ω is the angular frequency, t is the time in seconds, and φ is the phase angle.

Given that the weight is pulled down 4 inches below the equilibrium position, we can determine the amplitude of the motion. Since the weight is pulled down, the amplitude is the distance from the equilibrium to the maximum displacement. In this case, the amplitude would be 4 inches. Therefore, A = 4.

We are also given that the frequency of the system is 8/π cycles per second. The angular frequency (ω) is related to the frequency (f) by the formula ω = 2πf. So, in this case, the angular frequency (ω) would be 2π * (8/π) = 16 radians per second.

To determine the phase angle (φ), we need some additional information. If we are given the initial position or velocity of the weight, we can determine the phase angle. However, the question does not provide us with this information. Therefore, we assume that at time t = 0, the weight is at its equilibrium position. This implies that the phase angle (φ) should be zero.

Therefore, the trigonometric model that gives the position of the weight at time t seconds is:

x(t) = 4 * cos(16t)