Complete the table.
\begin{tabular}{|c|c|c|}
\hline \begin{tabular}{c}
Time \\
$( s )$
\end{tabular} & \begin{tabular}{c}
Distance \\
$( m )$
\end{tabular} & \begin{tabular}{c}
Height \\
$( m )$
\end{tabular} \\
\hline 0 & 0 & 0 \\
\hline$\frac{\pi}{2}$ & $a$ & $d$ \\
\hline$\pi$ & $b$ & $e$ \\
\hline$\frac{3 \pi}{2}$ & $c$ & $f$ \\
\hline $2 \pi$ & $2 \pi$ & 0 \\
\hline
\end{tabular}
Heights:
$d=\square e=\square f=\square$
Distances:
$a=\square \pi, b=\square \pi, c=\square \pi$
Answer (2)
To complete the table, the heights calculated are d = 1 , e = 0 , and f = − 1 ; the distances are a = 2 1 , b = 1 , and c = 2 3 . This is based on the circular motion where distance equals time and height is determined by the sine function. Using these relationships, we derive the completed values for each section of the table.
;
The distance traveled is the arc length, which is t since the radius is 1.
The height is given by sin ( t ) .
Calculate a = 2 π ⟹ a = 2 1 , b = π ⟹ b = 1 , c = 2 3 π ⟹ c = 2 3 .
Calculate d = sin ( 2 π ) = 1 , e = sin ( π ) = 0 , f = sin ( 2 3 π ) = − 1 . The final answers are d = 1 , e = 0 , f = − 1 , a = 2 1 , b = 1 , c = 2 3 .
Explanation
Understanding the Problem We are given a table relating time, distance, and height. We need to complete the table by finding the values of a , b , c , d , e , and f . The table suggests a circular motion with a radius of 1, where the distance traveled is the arc length and the height is given by sin ( t ) .
Setting up the Equations The distance traveled is given by the arc length, which is r × t , where r is the radius and t is the time. Since the radius is 1, the distance traveled is simply t . The height is given by sin ( t ) .
Calculating a At time t = 2 π , the distance is a = 2 π . Therefore, a = 2 1 .
Calculating b At time t = π , the distance is b = π . Therefore, b = 1 .
Calculating c At time t = 2 3 π , the distance is c = 2 3 π . Therefore, c = 2 3 .
Calculating d At time t = 2 π , the height is d = sin ( 2 π ) = 1 .
Calculating e At time t = π , the height is e = sin ( π ) = 0 .
Calculating f At time t = 2 3 π , the height is f = sin ( 2 3 π ) = − 1 .
Final Answer Therefore, the values are: a = 2 1 , b = 1 , c = 2 3 , d = 1 , e = 0 , f = − 1 .
Examples
Imagine a point moving around a circle of radius 1 at a constant speed. The table describes the position of the point at different times. The distance represents how far the point has traveled along the circle's circumference, and the height represents the vertical position of the point. This concept is used in physics to describe simple harmonic motion, such as the motion of a pendulum or a mass on a spring. Understanding circular motion and trigonometric functions is crucial in many areas of science and engineering.
