jackson takes a marker and draws an arrow at the top of his yo - yo, pointing it toward the string when it…
jackson takes a marker and draws an arrow at the top of his yo - yo, pointing it toward the string when it is all wound up. the location of the arrow on the yo - yo can be represented by a cosine function. the function that represents the location of the arrow is f(x)=2cos(8πx)+2, where x represents time in seconds, and f(x) represents the vertical distance in inches that the arrow is from the lowest point on the yo - yo. what is the width of the yo - yo? 4 inches 8 inches 2 inches 8π inches
Answer
Explanation:
Step1: Recall the amplitude - diameter relationship
For a cosine function of the form $y = A\cos(Bx)+C$, the amplitude $A$ gives half of the vertical distance between the maximum and minimum values of the function. The diameter of the yo - yo is related to the vertical distance between the highest and lowest positions of the arrow on the yo - yo. The given function is $f(x)=2\cos(8\pi x)+2$. Here, the amplitude $A = 2$.
Step2: Calculate the diameter (width) of the yo - yo
The vertical distance between the highest and lowest points of the arrow's motion (which is the width of the yo - yo) is equal to the difference between the maximum and minimum values of the function $f(x)$. For a cosine function $y = A\cos(Bx)+C$, the maximum value is $y_{max}=A + C$ and the minimum value is $y_{min}=-A + C$. The difference $y_{max}-y_{min}=(A + C)-(-A + C)=2A$. Since $A = 2$, the width of the yo - yo is $2A=4$ inches.
Answer:
4 inches