Final answer:
The vertical distance 'H' from the Ferris wheel seat to the ground as a function of time 't' is given by H(t) = 50 * sin(2π * t / 30) + 60. To find the rate of change when H=55 feet, we compute the derivative of H(t) with respect to time and evaluate it at the time 't' when H equals 55 feet.
Step-by-step explanation:
Vertical Distance as a Function of Time
The vertical distance 'H' that the seat of a Ferris wheel is off the ground as a function of time 't' can be modeled using trigonometric functions. Since the wheel has a diameter of 100 feet and rises 10 feet off the ground, the radius of the wheel's circular motion is 50 feet, and the center of the wheel's circular path is 60 feet above the ground. Hence, we can use a sine function to express 'H'. If 't=0' corresponds to the seat being at the bottom, then:
H(t) = 50 * sin(2π * t / 30) + 60
Here, '2π' represents a full revolution in radian measure, and 't/30' accounts for the fact that one full revolution takes 30 seconds.
Rate of Change when H=55 feet
When the height 'H' is 55 feet, we want to find the rate of change of 'H' with respect to time. By using the derivative of 'H(t)' and setting 'H' to 55, we'll be able to solve for 'dh/dt', which is the rate at which the Ferris wheel seat is rising.
Calculating the Derivative
The derivative of 'H(t)' with respect to 't' gives us the velocity of the seat at any given time:
dh/dt = (2π / 30) * 50 * cos(2π * t / 30)
Now we need to solve for 't' when 'H=55', then we can plug that value into our expression for 'dh/dt' to find the rate of change at 'H=55' feet.