Question

Griffin is riding his bike down the street in Churchville, N.Y. at a constant speed, when a nail gets caught in one of his tires. The height of the nail above the ground, in inches, can be represented by the trigonometric function f(t)=-13cos(0.8pi(t))+13, where t represents the time (in seconds) since the nail first became caught in the tire. A) Determine the period of f(t). B) Interpret what the period represents in this context. C) On the grid below, graph at least one cycle of f(t) that includes the y-intercept of the function.

Answer 1

The period of the function f(t)=-13cos(0.8pi(t))+13 is 2.5 seconds. The period represents the time it takes for the height of the nail to complete one full cycle. To graph one cycle of f(t), substitute values for t in the function and plot the corresponding y values.

Step-by-step explanation:

A) The period of the function f(t)=-13cos(0.8pi(t))+13 can be found by dividing the period of the cosine function, which is 2π, by the coefficient of t in the argument of the cosine function. In this case, the coefficient is 0.8π, so the period of f(t) is 2π / 0.8π = 2.5 seconds.

B) The period represents the time it takes for the height of the nail above the ground to complete one full cycle. In this context, it represents the time it takes for the nail in Griffin's tire to go through one complete up-and-down motion due to the oscillation of the tire caused by the nail.

C) To graph one cycle of f(t), we can choose a range of t values that covers one full period. Let's choose t values from 0 to 2.5. Substituting these values into the function, we get the corresponding y values. The graph of one cycle of f(t) will have points connecting these y values.