The second hand of a clock is 4 inches long. Describe the height of the tip of the second hand from the bottom of a clock. (The radius of the clock is 1 inch more than the length of the second hand.)
The tip of the second hand travels on a 4" circle, inside a 5" circle. So, at t seconds into the minute, the tip of the second hand is at height
5+4cos(t/60 * 360)° = 5 + 4cos(6t)°
from the bottom of the clock.
To determine the height of the tip of the second hand from the bottom of the clock, we first need to find the radius of the clock.
Given that the second hand is 4 inches long and the radius of the clock is 1 inch more than the length of the second hand, we can calculate the radius.
Radius of the clock = Length of the second hand + 1 = 4 + 1 = 5 inches.
Now, let's consider the clock face as a coordinate plane, with the center of the clock at (0, 0). The bottom of the clock is located at the point (0, -5) since the radius of the clock is downwards from the center.
Since the second hand rotates around the center of the clock, it traces a circular path. The tip of the second hand can be represented as a point on the circumference of this circle.
The height of the tip of the second hand is the y-coordinate of this point on the circumference.
To find this y-coordinate, we can use trigonometry. Since the second hand is like the radius of the circle, the coordinates of the tip of the second hand can be calculated using the angle formed with respect to the positive x-axis.
Assuming the initial position of the second hand is at the positive x-axis (3 o'clock position), the angle formed by it can be represented as θ (theta).
Now, we can use the sine function to find the height of the tip:
sin(θ) = opposite/hypotenuse.
The opposite side of the right triangle formed by the angle θ will provide the y-coordinate of the point on the circumference.
So, sin(θ) = y-coordinate/Length of the second hand.
Rearranging the equation, we have y-coordinate = Length of the second hand * sin(θ).
Since the length of the second hand is 4 inches, we can substitute that value into the equation:
y-coordinate = 4 * sin(θ).
To complete the calculation, we need to know the angle θ at the specific moment we're interested in. Once we have that value, we can calculate the corresponding y-coordinate using the equation.
Therefore, to determine the height of the tip of the second hand from the bottom of a clock, we need to know the current angle θ. Without that information, we cannot provide an exact answer.