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Using arc 1(A) & arc 2(B) and difference in the two arc radius is W, Find r1, r2 ?

Guest Aug 27, 2017
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#1
+1366
+1

This diagram has a lot of information, and it can be somewhat difficult to decipher. With 2 radii, it should be possible to figure out the central angle. Before we start, we must understand arc length and its formula. The formula is below

$$\text{Arclength}=\frac{m^{\circ}}{360^{\circ}}*2\pi r$$

m = measure of the central angle, in degrees

r = radius of the circle

First, let's create 2 equations that demonstrate the relationship of r1 in this diagram:

 $$l=\frac{m^{\circ}}{360^{\circ}}*2\pi r$$ I am using "l" as my chosen variable for arc length. Of course, we already know the arc length; it is A, or 32. And we know that arc length corresponds with r1. $$32=\frac{m^{\circ}}{360^{\circ}}*2\pi r_1$$ Great! Now, I will solve for r1 now. First, let's simplify the right hand side of the equation. $$\frac{m^{\circ}}{360^{\circ}}*\frac{2\pi r_1}{1}$$ Multiply the fractions together. $$\frac{2\pi r_1 m^{\circ}}{360^{\circ}}$$ The numerator and denominator have a GCF of 2, so factor that out. $$\frac{\pi r_1 m^{\circ}}{180^{\circ}}$$ Reinsert this into the orginal equation. $$32=\frac{\pi r_1 m^{\circ}}{180^{\circ}}$$ Multiply by 180 on both sides of the equation. $$5760^{\circ}=\pi r_1 m^{\circ}$$ Divide by $$\pi m^{\circ}$$ to isolate r1 $$r_1=\frac{5760^{\circ}}{\pi m^{\circ}}$$

Let's do the exact same process for r2

 $$34=\frac{m^{\circ}}{360^{\circ}}*2\pi r_2$$ We already from before what the right hand side simplifies to, so let's do that. $$34=\frac{\pi r_2 m^{\circ}}{180^{\circ}}$$ Do the same process to solve for r2. Multiply both sides by 180. $$6120^{\circ}=\pi r_2 m^{\circ}$$ Divide both sides by $$\pi m^{\circ}$$. $$r_2=\frac{6120^{\circ}}{\pi m^{\circ}}$$

We have two equations, and they are the following:

{

$$r_2=\frac{6120^{\circ}}{\pi m^{\circ}}$$

$$r_1=\frac{5760^{\circ}}{\pi m^{\circ}}$$

{

Now, let's subtract the 2 equations from each other. Let's see what happens. Unfortuntaely, I am no genius with LaTeX, so I could not line up the equal signs. Hopefully, you get the point...

$$\begin{eqnarray*} r_2=\frac{6120^{\circ}}{\pi m^{\circ}}\\ -\left(r_1=\frac{5760^{\circ}}{\pi m^{\circ}}\right)\\ \end{eqnarray*}$$

Let's subtract the two equations together. Let's start with the left hand side subtraction. $$r_2-r_1=1.5$$, according to the given info. $$\frac{6120^{\circ}}{\pi m^{\circ}}-\frac{5760}{\pi m^{\circ}}=\frac{360}{\pi m^{\circ}}$$. Knowing this, we can now solve for the central angle.

 $$1.5=\frac{360}{\pi m^{\circ}}$$ Multiply by $$\pi m^{\circ}$$ on both sides. $$1.5\pi m^{\circ}={360}$$ Divide by $$1.5\pi$$ $$m^{\circ}=\frac{360}{1.5\pi}$$ We can simplify the right hand side further, actually. First, let's convert 1.5 into a fraction. $$\frac{360}{1.5\pi}=\frac{360}{\frac{3\pi}{2}}$$ In a fraction, $$\frac{a}{\frac{b}{c}}=\frac{a*c}{b}$$. Let's apply it. $$\frac{360}{\frac{3\pi}{2}}=\frac{360*2}{3\pi}$$ In the numerator and denominator, 3 can be factored out. $$m^{\circ}=\frac{120*2}{\pi}=\frac{240}{\pi}$$

Now that I know the central angle in its exact form, I can now substitute it back into the arc length formula to ge the measure of the radii. Let's do that! FIrst, I'll solve for r1 first.

 $$l=\frac{m^{\circ}}{360^{\circ}}*2\pi r$$ As a reminder, this is the formula for arc length. Substitute the values we already know. $$32=\frac{m^{\circ}}{360}*2\pi r_1$$ I have decided to leave out the measure of the angle for now, so the equation does not look like a monstrosity. Multiply both sides by 360. $$11520=2\pi r_1m^{\circ}$$ Divide by 2 on both sides. $$5760=\pi r_1m^{\circ}$$ Now, let's plug in m. $$5760=\pi r_1\left(\frac{240}{\pi}\right)$$ Eliminate the pi's as they cancel out. $$5760=240r_1$$ Divide by 240 on both sides. $$r_1=\frac{5760}{240}=24units$$ I happen to know that 24^2=576, so this division was relatively easy for me. I will not forget the units, either!

We don't have to do this same process for r2 luckily as $$r_1+1.5=r_2$$. Therefore, $$r_2=24+1.5=25.5units$$

TheXSquaredFactor  Aug 27, 2017
#2
+78678
+1

Note that the formula for the arc length, S,  is

S = r * theta       where  "theta" is the central angle forming the arc and r is the radius

And let   r  = "r1"

So....we have this system

32  = r * theta     →   32/r  = theta    (1)        and  letting r2 = r + 1.5

34  =  (r + 1.5) * theta         (2)

Sub (1) into (2)   for   "theta"    and we have that

34  = (r + 1.5) (32/r)     multiply through by r

34r =  (r + 1.5) (32)     simplify

34r  = 32r + 48      subtract 32r  from both sides

2r  = 48       divide both sides by 2

r  = 24      =   " r1 "

And

r + 1.5  =  24 + 1.5  = 25.5  =  "r2"

CPhill  Aug 27, 2017
edited by CPhill  Aug 27, 2017
edited by CPhill  Aug 27, 2017

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