A circle has a chord that goes from #pi/12 # to #pi/4 # radians on the circle. If the area of the circle is #32 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #pi/12  # to #pi/4  # radians on the circle.  If the area of the circle is #32   pi  #, what is the length of the chord?

Theodore Abad · Accepted Answer

length of chord #=4(sqrt3-1)~~2.928#

As area of a circle is given by #pir^2#, and it is #32pi#, we have #r=sqrt32#.

As shown in the figure, the angle #Theta# subtended by the chord at the centre is :
#Theta=pi/4-pi/12=pi/6#
#=> Theta/2=pi/12#

#=> AM=rsin(Theta/2)#

#=># length of chord #AB=2AM=2*r*sin(Theta/2)#
#= 2*sqrt32*sin(pi/12)~~2.928#

exact value:
#AB=2*sqrt32*((sqrt6-sqrt2)/4)#
#=1/2(8sqrt3-8)#
=#4(sqrt3-1)#

Violet Aldrich · Answer

undefined To find the length of the chord, we first need to determine the radius of the circle. We can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.

Given that the area of the circle is 32π, we can solve for the radius:

32π = πr^2

Divide both sides by π:

32 = r^2

Take the square root of both sides to find the radius:

r = √32 = 4√2

Now that we have the radius, we can find the length of the chord using trigonometry. The chord divides the circle into two segments, each with a central angle of (π/4) - (π/12) = π/6 radians.

The formula to find the length of the chord in terms of the radius (r) and the central angle (θ) is:

Chord length = 2 * r * sin(θ/2)

Substituting the values, we get:

Chord length = 2 * (4√2) * sin(π/12)

Now, we can calculate the value of sin(π/12) using a calculator. Once we have that value, we can plug it into the equation to find the length of the chord.