The vertical location of the tip of a clock’s second hand (y) relative to the clock’s center measured in centimetres is modelled by the equation y= 3cos (πs/30), where s is the number of seconds that have passed.
a) How long is the second hand, and where does it start?
b) Use transformations to explain how you could change the original equation to find the vertical location of the tip of the hour and minute hands in terms of y and s if the hour hand is 1 centimetre long and the minute hand is 4 centimetres long.
c) What is a simple way you could change the equations to represent the hands’ horizontal positions? Why does this work?
(a) 3
(b) change the period. Note that cos(2ks) has a period 1/2 as long as cos(ks)
(c) cos(k(s-θ)) is shifted later by θ
a) The length of the second hand is not given in the equation. However, the equation only models the vertical position of the tip of the second hand. It does not give information about the length of the second hand. The equation y = 3cos(πs/30) indicates the vertical displacement of the second hand at different seconds.
To determine where the second hand starts, we need to substitute s = 0 into the equation.
y = 3cos(π(0)/30)
y = 3cos(0)
y = 3
So, at the starting position, the tip of the second hand is 3 centimeters above or below the center of the clock.
b) To find the vertical location of the tip of the hour and minute hands, we can use similar equations to model their positions. The equation for the hour hand can be:
y_h = yc + ah*cos(πs_h/180)
Here, yc represents the vertical position of the center of the clock, and ah represents the length of the hour hand. s_h represents the number of seconds that have passed for the hour hand.
Similarly, the equation for the minute hand can be:
y_m = yc + am*cos(πs_m/180)
Here, am represents the length of the minute hand, and s_m represents the number of seconds that have passed for the minute hand.
c) To represent the hands' horizontal positions, we can use similar equations with the sine function instead of the cosine function. The equation for the second hand would become:
x = xc + as*sin(πs/30)
Here, xc represents the horizontal position of the center of the clock, and as represents the length of the second hand. s represents the number of seconds that have passed.
This works because the cosine function models the vertical displacement, while the sine function models the horizontal displacement in a circular motion. By substituting sine instead of cosine in the equations, we can determine the horizontal positions of the clock hands.
a) To find how long the second hand is, we need to analyze the equation y = 3cos(πs/30). In this equation, y represents the vertical location of the tip of the second hand, and s represents the number of seconds that have passed.
Since the equation contains the term "cos(πs/30)", we know that the value of y will vary between -3 and 3. This means that the second hand is 6 centimeters long, with its tip oscillating vertically between -3 and 3 centimeters.
To determine where the second hand starts, we can look at the equation when s equals zero. Plugging in s = 0 into the equation, we get y = 3cos(0) = 3. This means that the tip of the second hand starts 3 centimeters above the clock's center.
b) To find the vertical location of the hour and minute hands, we can use transformations on the original equation y = 3cos(πs/30). Here's how we can modify the equation to represent the vertical location of the different hands:
1. Hour hand (1 centimeter long): To represent the hour hand, we can scale down the amplitude of the original equation. Since the length of the hour hand is smaller, we need to reduce the oscillation range. The modified equation could be y_hour = a*cos(πs/30), where 'a' represents the amplitude. For the hour hand, 'a' can be set to 1.
2. Minute hand (4 centimeters long): Similar to the hour hand, we can scale down the amplitude to represent the minute hand. The modified equation could be y_minute = b*cos(πs/30), where 'b' represents the amplitude. For the minute hand, 'b' can be set to 4.
By substituting the appropriate values for 'a' and 'b' into the corresponding modified equations, we can determine the vertical locations of the hour and minute hands at any given time.
c) To represent the hands' horizontal positions, we need to modify the equations. The cosine function is responsible for the vertical oscillation, so we need to change it to a function that oscillates horizontally. The sine function is suitable for this purpose.
To represent the horizontal position of the hands, we can modify the equations as follows:
1. Hour hand: x_hour = c*sin(πs/30), where 'c' is a scaling factor for the horizontal position of the hour hand.
2. Minute hand: x_minute = d*sin(πs/30), where 'd' is a scaling factor for the horizontal position of the minute hand.
By substituting the appropriate values for 'c' and 'd' into the corresponding modified equations, we can determine the horizontal positions of the hour and minute hands at any given time. This works because the sine function oscillates horizontally while maintaining a similar periodic nature to the cosine function.