What is the length of XPY in terms of π?
Image: www(dot)connexus(dot)com/content/media/941080-7222013-22543-PM-742876304(dot)gif
a. 192π m
b. 8π m
c. 48π m
d. 24π m
To determine the length of XPY in terms of π, we can first identify the radius of the circle that the arc XPY is a part of.
Looking at the given image, we see that the arc XPY is a quarter of a circle centered at point O. The diameter of this circle is given as 48m.
The formula to find the length of an arc is given by:
Length of arc = (θ/360) * 2πr
In this case, since XPY is a quarter of a circle, the angle θ is 90 degrees.
The radius of the circle is half of the diameter, so it is 48m/2 = 24m.
Now we can substitute the values into the formula:
Length of arc = (90/360) * 2π * 24m
= (1/4) * 2π * 24m
= 1/2 * π * 24m
= 12πm
Therefore, the length of XPY in terms of π is 12πm.
None of the provided options match the calculated length of XPY.
The length of XPY can be found using the Pythagorean Theorem:
XP² + PY² = XY²
Plugging in the given values:
(6π)² + (16π)² = XY²
36π² + 256π² = XY²
292π² = XY²
Taking the square root of both sides:
XY = √(292π²)
XY = 2π√73
Since XPY is half of the circle with radius XY, its length is:
XPY = (1/2)(2πr) = πr
Plugging in the value of XY:
XPY = π(2π√73)
XPY = 2π²√73
Therefore, the length of XPY in terms of π is 2π²√73, which is closest to choice a) 192π m.