Question

A 10-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 5 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The wheel rotates counterclockwise at 2.5 rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture. (a) Express the x and y coordinates of point A as functions of t:

x= and y= .
(b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds:

Incorrect: Your answer is incorrect.

(c) Express the x-coordinate of the right end of the rod at point B as a function of t:
Incorrect: Your answer is incorrect.
.
(d) Express the slope of the rod as a function of t:
Incorrect: Your answer is incorrect.
.
(c) Find the first time when the rod is tangent to the circle:
ur answer is incorrect.
seconds.
(d) At the time in (c), what is the slope of the rod?
.
(e) Find the second time when the rod is tangent to the circle:
(f) At the time in (e), what is the slope of the rod?

Answer 1

(a)The x and y coordinates of point A as functions of time t are given by:

x_t = 5cos(2.5t)

y_t = 5sin(2.5t)

(b)The formula for the slope of the tangent line to the circle at point A at time t seconds is obtained by taking the derivative of the y-coordinate with respect to the x-coordinate:

(dy/dx)_t = -tan(2.5t)

(c)The x-coordinate of the right end of the rod at point B as a function of time t is given by:

x_B_t = 10cos(2.5t)

Step-by-step explanation:

(a)

The position of point A is determined by the trigonometric functions representing the x and y coordinates in terms of time t. Since the wheel is rotating counterclockwise at a rate of 2.5 revolutions per second, the argument of the trigonometric functions is 2.5t, and the radius of the wheel is 5 cm.

(b)

To find the slope of the tangent line, we take the derivative of the y-coordinate with respect to the x-coordinate. The tangent function naturally arises, resulting in the formula (dy/dx)_t = -tan(2.5t).

(c)

The x-coordinate of point B is determined by the same trigonometric function as point A, but the radius is doubled, resulting in x_B_t = 10cos(2.5t).

In conclusion, the positions of points A and B on the rotating system are expressed as functions of time using trigonometric functions. The slope of the tangent line and the x-coordinate of point B are derived accordingly, providing a comprehensive understanding of the rod's movement on the wheel.