The height, [tex]$h$[/tex], in feet of the tip of the minute hand of a wall clock as a function of time, [tex]$t$[/tex], in minutes can be modeled by the equation [tex]$h=0.75 \cos \left(\frac{\pi}{30}(t-15)\right)+8$[/tex]. Which number (from 1 to 12 ) is the minute hand pointing to at [tex]$t=0$[/tex] ?
A. 3
B. 6
C. 9
D. 12
Answer (1)
Substitute t = 0 into the equation: h = 0.75 ", "cos ", "left(", "frac{", "pi}{30}(0-15)\right)+8 .
Simplify the expression: h = 0.75 ", "cos ", "left(-", "frac{", "pi}{2}\right)+8 .
Calculate the cosine: $
Explanation
Understanding the Problem We are given the equation for the height of the tip of the minute hand as a function of time: h=0.75 ", "cos ", "left(", "frac{", "pi}{30}(t-15)\right)+8 . We want to find the position of the minute hand at t = 0 . This means we need to find the height h when t = 0 .
Substituting t=0 Substitute t = 0 into the equation: h = 0.75 ", "cos ", "left(", "frac{", "pi}{30}(0-15)\right)+8 = 0.75 ", "cos ", "left(-", "frac{", "pi}{2}\right)+8
Calculating the Cosine We know that $
Finding the Height Now, substitute this value back into the equation for h :
h = 0.75 ( 0 ) + 8 = 0 + 8 = 8
Determining the Number The height h = 8 corresponds to the minute hand pointing directly to the right. On a clock, the number 3 is located on the right side. Therefore, the minute hand is pointing to the number 3 at t = 0 .
Examples
Understanding the position of clock hands can be related to understanding cyclical patterns in various real-world scenarios, such as tidal movements, seasonal changes, or even scheduling tasks throughout the day. For example, if you need to schedule a meeting at a specific time, knowing the position of the minute hand helps you visualize the time and plan accordingly. This problem demonstrates how trigonometric functions can model periodic phenomena and aid in time management and planning.
