I'm having a lot of difficulty with getting this to make sense and the answer in the book is just '8.4 in'
Q " You want to make an 80 degree angle by marking an arc oin the perimeter of a 12-in. diameter disk and drawing lines from the ends of the arc to the disk's center. To the nearest tenth of an inch, how long should the arc be?"
I tried going about it 2 different ways…
I converted 80 degrees into radians in order to use the special property that theta in radians = arclength/radius or: $$ \frac{80}{180}\pi = \frac{4}{9}\pi $$
And by $$\theta=\frac{s}{r},$$
$$\frac{4}{9}\pi=\frac{s}{6}$$
So, $$s = \frac{24}{9}\pi $$
But why?! Why then does my book say 8.4 inches!
So then I tried going about it with my stupid head instead of the book… and I thought OK well a circle is 360 degrees.. lets see if I can work it out that way
$$\frac{80}{360}=\frac{2}{9}$$
So 2/9's of the distance around the circle which I know to be:$$2r\pi = 12\pi$$ inches is:$$\frac{24}{9}\pi = \frac{24}{9}*3.14 = ~~8.37$$
Best Answer
$$\frac{24}{9}\pi\; \text{ inches} \approx 8.3775804097\;\text{inches} \approx 8.4 \;\text{inches} $$ And for your second method it should give the same as your first: $$\frac{80}{360}=\frac{2}{9} \rightarrow \frac{2}{9}\times (2r\pi)=\frac{2}{9}\times (12\pi)=\frac{24\pi}{9}$$