An ant is crawling along the x-axis such that the graph of its position on the x-axis
versus time is a semi-circle (see figure). The total distance covered in the 4 s is
What figure?
Just subtract the starting position from the ending position to get the distance traveled...
To calculate the total distance covered by the ant in 4 seconds, we need to find the length of the semicircle formed by the graph of its position on the x-axis versus time.
Given that the graph is a semicircle, we can imagine it as the upper half of a complete circle, where the radius of the circle represents the distance covered by the ant.
To find the radius of the circle, we need to know the maximum position of the ant on the x-axis. Let's assume that the maximum position is denoted by the coordinate (x_max, y_max).
To find the maximum position of the ant, we need to consider the equation of the semicircle graph. The equation is given by:
x^2 + y^2 = r^2
where x represents time, y represents the position of the ant, and r represents the radius of the circle.
Since the graph is a semicircle only on the positive side of the x-axis, we can rewrite the equation as:
x^2 + y^2 = r^2, for x >= 0
Now let's consider the time range of 4 seconds, denoted as t [0, 4]. To find the maximum position of the ant, we need to evaluate the equation at t = 4:
4^2 + y^2 = r^2
Since we're interested in the distance covered, we only need the absolute value of y. Let's assume the absolute value of y at t = 4 is denoted by d_max.
Now, to calculate the total distance covered by the ant in 4 seconds, we need to find the circumference of the circle formed by the semicircle graph. The circumference is given by:
C = πd
where d represents the diameter of the circle.
Since the diameter is twice the radius, we have:
C = 2πr
Substituting the value of d_max into the formula, we get:
C = 2π * d_max
Therefore, to find the total distance covered by the ant in 4 seconds, you need to calculate 2π times the absolute value of y at t = 4.